:: Tools for Computing Prime Numbers - Part I
:: by Library Committee
::
:: Received October 31, 2022
:: Copyright (c) 2022-2025 Association of Mizar Users

theorem Th1: :: XPRIMET1:1

for p being Prime st p < 3 holds
p = 2

proof end;

theorem :: XPRIMET1:2

for k being Nat
for p being Prime st p * p <= k & k < 9 holds
p = 2

proof end;

theorem Ttool5a: :: XPRIMET1:3

for p being Prime holds
( not p < 5 or p = 2 or p = 3 )

proof end;

theorem :: XPRIMET1:4

for k being Nat
for p being Prime st p * p <= k & k < 25 & not p = 2 holds
p = 3

proof end;

theorem Ttool7a: :: XPRIMET1:5

for p being Prime holds
( not p < 7 or p = 2 or p = 3 or p = 5 )

proof end;

theorem :: XPRIMET1:6

for k being Nat
for p being Prime st p * p <= k & k < 49 & not p = 2 & not p = 3 holds
p = 5

proof end;

theorem Ttool11a: :: XPRIMET1:7

for p being Prime holds
( not p < 11 or p = 2 or p = 3 or p = 5 or p = 7 )

proof end;

theorem :: XPRIMET1:8

for k being Nat
for p being Prime st p * p <= k & k < 121 & not p = 2 & not p = 3 & not p = 5 holds
p = 7

proof end;

theorem Ttool13a: :: XPRIMET1:9

for p being Prime holds
( not p < 13 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 )

proof end;

theorem :: XPRIMET1:10

for k being Nat
for p being Prime st p * p <= k & k < 169 & not p = 2 & not p = 3 & not p = 5 & not p = 7 holds
p = 11

proof end;

theorem Ttool17a: :: XPRIMET1:11

for p being Prime holds
( not p < 17 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 )

proof end;

theorem :: XPRIMET1:12

for k being Nat
for p being Prime st p * p <= k & k < 289 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 holds
p = 13

proof end;

theorem Ttool19a: :: XPRIMET1:13

for p being Prime holds
( not p < 19 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 )

proof end;

theorem :: XPRIMET1:14

for k being Nat
for p being Prime st p * p <= k & k < 361 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 holds
p = 17

proof end;

theorem Ttool23a: :: XPRIMET1:15

for p being Prime holds
( not p < 23 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 )

proof end;

theorem :: XPRIMET1:16

for k being Nat
for p being Prime st p * p <= k & k < 529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 holds
p = 19

proof end;

theorem Ttool29a: :: XPRIMET1:17

for p being Prime holds
( not p < 29 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 )

proof end;

theorem :: XPRIMET1:18

for k being Nat
for p being Prime st p * p <= k & k < 841 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 holds
p = 23

proof end;

theorem Ttool31a: :: XPRIMET1:19

for p being Prime holds
( not p < 31 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 )

proof end;

theorem :: XPRIMET1:20

for k being Nat
for p being Prime st p * p <= k & k < 961 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 holds
p = 29

proof end;

theorem Ttool37a: :: XPRIMET1:21

for p being Prime holds
( not p < 37 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 )

proof end;

theorem :: XPRIMET1:22

for k being Nat
for p being Prime st p * p <= k & k < 1369 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 holds
p = 31

proof end;

theorem Ttool41a: :: XPRIMET1:23

for p being Prime holds
( not p < 41 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 )

proof end;

theorem :: XPRIMET1:24

for k being Nat
for p being Prime st p * p <= k & k < 1681 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 holds
p = 37

proof end;

theorem Ttool43a: :: XPRIMET1:25

for p being Prime holds
( not p < 43 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 )

proof end;

theorem :: XPRIMET1:26

for k being Nat
for p being Prime st p * p <= k & k < 1849 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 holds
p = 41

proof end;

theorem Ttool47a: :: XPRIMET1:27

for p being Prime holds
( not p < 47 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 )

proof end;

theorem :: XPRIMET1:28

for k being Nat
for p being Prime st p * p <= k & k < 2209 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 holds
p = 43

proof end;

theorem Ttool53a: :: XPRIMET1:29

for p being Prime holds
( not p < 53 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 )

proof end;

theorem :: XPRIMET1:30

for k being Nat
for p being Prime st p * p <= k & k < 2809 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 holds
p = 47

proof end;

theorem Ttool59a: :: XPRIMET1:31

for p being Prime holds
( not p < 59 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 )

proof end;

theorem :: XPRIMET1:32

for k being Nat
for p being Prime st p * p <= k & k < 3481 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 holds
p = 53

proof end;

theorem Ttool61a: :: XPRIMET1:33

for p being Prime holds
( not p < 61 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 )

proof end;

theorem :: XPRIMET1:34

for k being Nat
for p being Prime st p * p <= k & k < 3721 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 holds
p = 59

proof end;

theorem Ttool67a: :: XPRIMET1:35

for p being Prime holds
( not p < 67 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 )

proof end;

theorem :: XPRIMET1:36

for k being Nat
for p being Prime st p * p <= k & k < 4489 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 holds
p = 61

proof end;

theorem Ttool71a: :: XPRIMET1:37

for p being Prime holds
( not p < 71 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 )

proof end;

theorem :: XPRIMET1:38

for k being Nat
for p being Prime st p * p <= k & k < 5041 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 holds
p = 67

proof end;

theorem Ttool73a: :: XPRIMET1:39

for p being Prime holds
( not p < 73 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 )

proof end;

theorem :: XPRIMET1:40

for k being Nat
for p being Prime st p * p <= k & k < 5329 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 holds
p = 71

proof end;

theorem Ttool79a: :: XPRIMET1:41

for p being Prime holds
( not p < 79 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 )

proof end;

theorem :: XPRIMET1:42

for k being Nat
for p being Prime st p * p <= k & k < 6241 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 holds
p = 73

proof end;

theorem Ttool83a: :: XPRIMET1:43

for p being Prime holds
( not p < 83 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 )

proof end;

theorem :: XPRIMET1:44

for k being Nat
for p being Prime st p * p <= k & k < 6889 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 holds
p = 79

proof end;

theorem Ttool89a: :: XPRIMET1:45

for p being Prime holds
( not p < 89 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 )

proof end;

theorem :: XPRIMET1:46

for k being Nat
for p being Prime st p * p <= k & k < 7921 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 holds
p = 83

proof end;

theorem Ttool97a: :: XPRIMET1:47

for p being Prime holds
( not p < 97 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 )

proof end;

theorem :: XPRIMET1:48

for k being Nat
for p being Prime st p * p <= k & k < 9409 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 holds
p = 89

proof end;

theorem Ttool101a: :: XPRIMET1:49

for p being Prime holds
( not p < 101 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 )

proof end;

theorem :: XPRIMET1:50

for k being Nat
for p being Prime st p * p <= k & k < 10201 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 holds
p = 97

proof end;

theorem Ttool103a: :: XPRIMET1:51

for p being Prime holds
( not p < 103 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 )

proof end;

theorem :: XPRIMET1:52

for k being Nat
for p being Prime st p * p <= k & k < 10609 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 holds
p = 101

proof end;

theorem Ttool107a: :: XPRIMET1:53

for p being Prime holds
( not p < 107 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 )

proof end;

theorem :: XPRIMET1:54

for k being Nat
for p being Prime st p * p <= k & k < 11449 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 holds
p = 103

proof end;

theorem Ttool109a: :: XPRIMET1:55

for p being Prime holds
( not p < 109 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 )

proof end;

theorem :: XPRIMET1:56

for k being Nat
for p being Prime st p * p <= k & k < 11881 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 holds
p = 107

proof end;

theorem Ttool113a: :: XPRIMET1:57

for p being Prime holds
( not p < 113 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 )

proof end;

theorem :: XPRIMET1:58

for k being Nat
for p being Prime st p * p <= k & k < 12769 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 holds
p = 109

proof end;

theorem Ttool127a: :: XPRIMET1:59

for p being Prime holds
( not p < 127 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 )

proof end;

theorem :: XPRIMET1:60

for k being Nat
for p being Prime st p * p <= k & k < 16129 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 holds
p = 113

proof end;

theorem Ttool131a: :: XPRIMET1:61

for p being Prime holds
( not p < 131 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 )

proof end;

theorem :: XPRIMET1:62

for k being Nat
for p being Prime st p * p <= k & k < 17161 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 holds
p = 127

proof end;

theorem Ttool137a: :: XPRIMET1:63

for p being Prime holds
( not p < 137 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 )

proof end;

theorem :: XPRIMET1:64

for k being Nat
for p being Prime st p * p <= k & k < 18769 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 holds
p = 131

proof end;

theorem Ttool139a: :: XPRIMET1:65

for p being Prime holds
( not p < 139 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 )

proof end;

theorem :: XPRIMET1:66

for k being Nat
for p being Prime st p * p <= k & k < 19321 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 holds
p = 137

proof end;

theorem Ttool149a: :: XPRIMET1:67

for p being Prime holds
( not p < 149 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 )

proof end;

theorem :: XPRIMET1:68

for k being Nat
for p being Prime st p * p <= k & k < 22201 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 holds
p = 139

proof end;

theorem Ttool151a: :: XPRIMET1:69

for p being Prime holds
( not p < 151 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 )

proof end;

theorem :: XPRIMET1:70

for k being Nat
for p being Prime st p * p <= k & k < 22801 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 holds
p = 149

proof end;

theorem Ttool157a: :: XPRIMET1:71

for p being Prime holds
( not p < 157 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 )

proof end;

theorem :: XPRIMET1:72

for k being Nat
for p being Prime st p * p <= k & k < 24649 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 holds
p = 151

proof end;

theorem Ttool163a: :: XPRIMET1:73

for p being Prime holds
( not p < 163 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 )

proof end;

theorem :: XPRIMET1:74

for k being Nat
for p being Prime st p * p <= k & k < 26569 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 holds
p = 157

proof end;

theorem Ttool167a: :: XPRIMET1:75

for p being Prime holds
( not p < 167 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 )

proof end;

theorem :: XPRIMET1:76

for k being Nat
for p being Prime st p * p <= k & k < 27889 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 holds
p = 163

proof end;

theorem Ttool173a: :: XPRIMET1:77

for p being Prime holds
( not p < 173 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 )

proof end;

theorem :: XPRIMET1:78

for k being Nat
for p being Prime st p * p <= k & k < 29929 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 holds
p = 167

proof end;

theorem Ttool179a: :: XPRIMET1:79

for p being Prime holds
( not p < 179 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 )

proof end;

theorem :: XPRIMET1:80

for k being Nat
for p being Prime st p * p <= k & k < 32041 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 holds
p = 173

proof end;

theorem Ttool181a: :: XPRIMET1:81

for p being Prime holds
( not p < 181 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 )

proof end;

theorem :: XPRIMET1:82

for k being Nat
for p being Prime st p * p <= k & k < 32761 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 holds
p = 179

proof end;

theorem Ttool191a: :: XPRIMET1:83

for p being Prime holds
( not p < 191 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 )

proof end;

theorem :: XPRIMET1:84

for k being Nat
for p being Prime st p * p <= k & k < 36481 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 holds
p = 181

proof end;

theorem Ttool193a: :: XPRIMET1:85

for p being Prime holds
( not p < 193 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 )

proof end;

theorem :: XPRIMET1:86

for k being Nat
for p being Prime st p * p <= k & k < 37249 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 holds
p = 191

proof end;

theorem Ttool197a: :: XPRIMET1:87

for p being Prime holds
( not p < 197 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 )

proof end;

theorem :: XPRIMET1:88

for k being Nat
for p being Prime st p * p <= k & k < 38809 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 holds
p = 193

proof end;

theorem Ttool199a: :: XPRIMET1:89

for p being Prime holds
( not p < 199 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 )

proof end;

theorem :: XPRIMET1:90

for k being Nat
for p being Prime st p * p <= k & k < 39601 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 holds
p = 197

proof end;

theorem Ttool211a: :: XPRIMET1:91

for p being Prime holds
( not p < 211 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 )

proof end;

theorem :: XPRIMET1:92

for k being Nat
for p being Prime st p * p <= k & k < 44521 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 holds
p = 199

proof end;

theorem Ttool223a: :: XPRIMET1:93

for p being Prime holds
( not p < 223 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 )

proof end;

theorem :: XPRIMET1:94

for k being Nat
for p being Prime st p * p <= k & k < 49729 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 holds
p = 211

proof end;

theorem Ttool227a: :: XPRIMET1:95

for p being Prime holds
( not p < 227 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 )

proof end;

theorem :: XPRIMET1:96

for k being Nat
for p being Prime st p * p <= k & k < 51529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 holds
p = 223

proof end;

theorem Ttool229a: :: XPRIMET1:97

for p being Prime holds
( not p < 229 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 )

proof end;

theorem :: XPRIMET1:98

for k being Nat
for p being Prime st p * p <= k & k < 52441 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 holds
p = 227

proof end;

theorem Ttool233a: :: XPRIMET1:99

for p being Prime holds
( not p < 233 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 )

proof end;

theorem :: XPRIMET1:100

for k being Nat
for p being Prime st p * p <= k & k < 54289 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 holds
p = 229

proof end;

theorem Ttool239a: :: XPRIMET1:101

for p being Prime holds
( not p < 239 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 )

proof end;

theorem :: XPRIMET1:102

for k being Nat
for p being Prime st p * p <= k & k < 57121 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 holds
p = 233

proof end;

theorem Ttool241a: :: XPRIMET1:103

for p being Prime holds
( not p < 241 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 )

proof end;

theorem :: XPRIMET1:104

for k being Nat
for p being Prime st p * p <= k & k < 58081 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 holds
p = 239

proof end;

theorem Ttool251a: :: XPRIMET1:105

for p being Prime holds
( not p < 251 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 )

proof end;

theorem :: XPRIMET1:106

for k being Nat
for p being Prime st p * p <= k & k < 63001 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 holds
p = 241

proof end;

theorem Ttool257a: :: XPRIMET1:107

for p being Prime holds
( not p < 257 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 )

proof end;

theorem :: XPRIMET1:108

for k being Nat
for p being Prime st p * p <= k & k < 66049 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 holds
p = 251

proof end;

theorem Ttool263a: :: XPRIMET1:109

for p being Prime holds
( not p < 263 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 )

proof end;

theorem :: XPRIMET1:110

for k being Nat
for p being Prime st p * p <= k & k < 69169 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 holds
p = 257

proof end;

theorem Ttool269a: :: XPRIMET1:111

for p being Prime holds
( not p < 269 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 )

proof end;

theorem :: XPRIMET1:112

for k being Nat
for p being Prime st p * p <= k & k < 72361 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 holds
p = 263

proof end;

theorem Ttool271a: :: XPRIMET1:113

for p being Prime holds
( not p < 271 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 )

proof end;

theorem :: XPRIMET1:114

for k being Nat
for p being Prime st p * p <= k & k < 73441 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 holds
p = 269

proof end;

theorem Ttool277a: :: XPRIMET1:115

for p being Prime holds
( not p < 277 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 )

proof end;

theorem :: XPRIMET1:116

for k being Nat
for p being Prime st p * p <= k & k < 76729 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 holds
p = 271

proof end;

theorem Ttool281a: :: XPRIMET1:117

for p being Prime holds
( not p < 281 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 )

proof end;

theorem :: XPRIMET1:118

for k being Nat
for p being Prime st p * p <= k & k < 78961 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 holds
p = 277

proof end;

theorem Ttool283a: :: XPRIMET1:119

for p being Prime holds
( not p < 283 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 )

proof end;

theorem :: XPRIMET1:120

for k being Nat
for p being Prime st p * p <= k & k < 80089 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 holds
p = 281

proof end;

theorem Ttool293a: :: XPRIMET1:121

for p being Prime holds
( not p < 293 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 )

proof end;

theorem :: XPRIMET1:122

for k being Nat
for p being Prime st p * p <= k & k < 85849 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 holds
p = 283

proof end;

theorem Ttool307a: :: XPRIMET1:123

for p being Prime holds
( not p < 307 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 )

proof end;

theorem :: XPRIMET1:124

for k being Nat
for p being Prime st p * p <= k & k < 94249 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 holds
p = 293

proof end;

theorem Ttool311a: :: XPRIMET1:125

for p being Prime holds
( not p < 311 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 )

proof end;

theorem :: XPRIMET1:126

for k being Nat
for p being Prime st p * p <= k & k < 96721 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 holds
p = 307

proof end;

theorem Ttool313a: :: XPRIMET1:127

for p being Prime holds
( not p < 313 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 )

proof end;

theorem :: XPRIMET1:128

for k being Nat
for p being Prime st p * p <= k & k < 97969 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 holds
p = 311

proof end;

theorem Ttool317a: :: XPRIMET1:129

for p being Prime holds
( not p < 317 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 )

proof end;

theorem :: XPRIMET1:130

for k being Nat
for p being Prime st p * p <= k & k < 100489 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 holds
p = 313

proof end;

theorem Ttool331a: :: XPRIMET1:131

for p being Prime holds
( not p < 331 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 )

proof end;

theorem :: XPRIMET1:132

for k being Nat
for p being Prime st p * p <= k & k < 109561 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 holds
p = 317

proof end;

theorem Ttool337a: :: XPRIMET1:133

for p being Prime holds
( not p < 337 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 )

proof end;

theorem :: XPRIMET1:134

for k being Nat
for p being Prime st p * p <= k & k < 113569 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 holds
p = 331

proof end;

theorem Ttool347a: :: XPRIMET1:135

for p being Prime holds
( not p < 347 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 )

proof end;

theorem :: XPRIMET1:136

for k being Nat
for p being Prime st p * p <= k & k < 120409 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 holds
p = 337

proof end;

theorem Ttool349a: :: XPRIMET1:137

for p being Prime holds
( not p < 349 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 )

proof end;

theorem :: XPRIMET1:138

for k being Nat
for p being Prime st p * p <= k & k < 121801 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 holds
p = 347

proof end;

theorem Ttool353a: :: XPRIMET1:139

for p being Prime holds
( not p < 353 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 )

proof end;

theorem :: XPRIMET1:140

for k being Nat
for p being Prime st p * p <= k & k < 124609 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 holds
p = 349

proof end;

theorem Ttool359a: :: XPRIMET1:141

for p being Prime holds
( not p < 359 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 )

proof end;

theorem :: XPRIMET1:142

for k being Nat
for p being Prime st p * p <= k & k < 128881 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 holds
p = 353

proof end;

theorem Ttool367a: :: XPRIMET1:143

for p being Prime holds
( not p < 367 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 )

proof end;

theorem :: XPRIMET1:144

for k being Nat
for p being Prime st p * p <= k & k < 134689 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 holds
p = 359

proof end;

theorem Ttool373a: :: XPRIMET1:145

for p being Prime holds
( not p < 373 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 )

proof end;

theorem :: XPRIMET1:146

for k being Nat
for p being Prime st p * p <= k & k < 139129 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 holds
p = 367

proof end;

theorem Ttool379a: :: XPRIMET1:147

for p being Prime holds
( not p < 379 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 )

proof end;

theorem :: XPRIMET1:148

for k being Nat
for p being Prime st p * p <= k & k < 143641 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 holds
p = 373

proof end;

theorem Ttool383a: :: XPRIMET1:149

for p being Prime holds
( not p < 383 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 )

proof end;

theorem :: XPRIMET1:150

for k being Nat
for p being Prime st p * p <= k & k < 146689 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 holds
p = 379

proof end;

theorem Ttool389a: :: XPRIMET1:151

for p being Prime holds
( not p < 389 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 )

proof end;

theorem :: XPRIMET1:152

for k being Nat
for p being Prime st p * p <= k & k < 151321 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 holds
p = 383

proof end;

theorem Ttool397a: :: XPRIMET1:153

for p being Prime holds
( not p < 397 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 )

proof end;

theorem :: XPRIMET1:154

for k being Nat
for p being Prime st p * p <= k & k < 157609 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 holds
p = 389

proof end;

theorem Ttool401a: :: XPRIMET1:155

for p being Prime holds
( not p < 401 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 )

proof end;

theorem :: XPRIMET1:156

for k being Nat
for p being Prime st p * p <= k & k < 160801 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 holds
p = 397

proof end;

theorem Ttool409a: :: XPRIMET1:157

for p being Prime holds
( not p < 409 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 )

proof end;

theorem :: XPRIMET1:158

for k being Nat
for p being Prime st p * p <= k & k < 167281 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 holds
p = 401

proof end;

theorem Ttool419a: :: XPRIMET1:159

for p being Prime holds
( not p < 419 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 )

proof end;

theorem :: XPRIMET1:160

for k being Nat
for p being Prime st p * p <= k & k < 175561 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 holds
p = 409

proof end;

theorem Ttool421a: :: XPRIMET1:161

for p being Prime holds
( not p < 421 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 )

proof end;

theorem :: XPRIMET1:162

for k being Nat
for p being Prime st p * p <= k & k < 177241 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 holds
p = 419

proof end;

theorem Ttool431a: :: XPRIMET1:163

for p being Prime holds
( not p < 431 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 )

proof end;

theorem :: XPRIMET1:164

for k being Nat
for p being Prime st p * p <= k & k < 185761 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 holds
p = 421

proof end;

theorem Ttool433a: :: XPRIMET1:165

for p being Prime holds
( not p < 433 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 )

proof end;

theorem :: XPRIMET1:166

for k being Nat
for p being Prime st p * p <= k & k < 187489 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 holds
p = 431

proof end;

theorem Ttool439a: :: XPRIMET1:167

for p being Prime holds
( not p < 439 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 )

proof end;

theorem :: XPRIMET1:168

for k being Nat
for p being Prime st p * p <= k & k < 192721 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 holds
p = 433

proof end;

theorem Ttool443a: :: XPRIMET1:169

for p being Prime holds
( not p < 443 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 )

proof end;

theorem :: XPRIMET1:170

for k being Nat
for p being Prime st p * p <= k & k < 196249 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 holds
p = 439

proof end;

theorem Ttool449a: :: XPRIMET1:171

for p being Prime holds
( not p < 449 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 )

proof end;

theorem :: XPRIMET1:172

for k being Nat
for p being Prime st p * p <= k & k < 201601 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 holds
p = 443

proof end;

theorem Ttool457a: :: XPRIMET1:173

for p being Prime holds
( not p < 457 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 )

proof end;

theorem :: XPRIMET1:174

for k being Nat
for p being Prime st p * p <= k & k < 208849 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 holds
p = 449

proof end;

theorem Ttool461a: :: XPRIMET1:175

for p being Prime holds
( not p < 461 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 )

proof end;

theorem :: XPRIMET1:176

for k being Nat
for p being Prime st p * p <= k & k < 212521 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 holds
p = 457

proof end;

theorem Ttool463a: :: XPRIMET1:177

for p being Prime holds
( not p < 463 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 )

proof end;

theorem :: XPRIMET1:178

for k being Nat
for p being Prime st p * p <= k & k < 214369 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 holds
p = 461

proof end;

theorem Ttool467a: :: XPRIMET1:179

for p being Prime holds
( not p < 467 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 )

proof end;

theorem :: XPRIMET1:180

for k being Nat
for p being Prime st p * p <= k & k < 218089 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 holds
p = 463

proof end;

theorem Ttool479a: :: XPRIMET1:181

for p being Prime holds
( not p < 479 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 )

proof end;

theorem :: XPRIMET1:182

for k being Nat
for p being Prime st p * p <= k & k < 229441 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 holds
p = 467

proof end;

theorem Ttool487a: :: XPRIMET1:183

for p being Prime holds
( not p < 487 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 )

proof end;

theorem :: XPRIMET1:184

for k being Nat
for p being Prime st p * p <= k & k < 237169 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 holds
p = 479

proof end;

theorem Ttool491a: :: XPRIMET1:185

for p being Prime holds
( not p < 491 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 )

proof end;

theorem :: XPRIMET1:186

for k being Nat
for p being Prime st p * p <= k & k < 241081 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 holds
p = 487

proof end;

theorem Ttool499a: :: XPRIMET1:187

for p being Prime holds
( not p < 499 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 )

proof end;

theorem :: XPRIMET1:188

for k being Nat
for p being Prime st p * p <= k & k < 249001 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 holds
p = 491

proof end;

theorem Ttool503a: :: XPRIMET1:189

for p being Prime holds
( not p < 503 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 )

proof end;

theorem :: XPRIMET1:190

for k being Nat
for p being Prime st p * p <= k & k < 253009 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 holds
p = 499

proof end;

theorem Ttool509a: :: XPRIMET1:191

for p being Prime holds
( not p < 509 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 )

proof end;

theorem :: XPRIMET1:192

for k being Nat
for p being Prime st p * p <= k & k < 259081 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 holds
p = 503

proof end;

theorem Ttool521a: :: XPRIMET1:193

for p being Prime holds
( not p < 521 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 )

proof end;

theorem :: XPRIMET1:194

for k being Nat
for p being Prime st p * p <= k & k < 271441 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 holds
p = 509

proof end;

theorem Ttool523a: :: XPRIMET1:195

for p being Prime holds
( not p < 523 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 )

proof end;

theorem :: XPRIMET1:196

for k being Nat
for p being Prime st p * p <= k & k < 273529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 holds
p = 521

proof end;

theorem Ttool541a: :: XPRIMET1:197

for p being Prime holds
( not p < 541 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 )

proof end;

theorem :: XPRIMET1:198

for k being Nat
for p being Prime st p * p <= k & k < 292681 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 holds
p = 523

proof end;

theorem Ttool547a: :: XPRIMET1:199

for p being Prime holds
( not p < 547 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 )

proof end;

theorem :: XPRIMET1:200

for k being Nat
for p being Prime st p * p <= k & k < 299209 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 holds
p = 541

proof end;

theorem Ttool557a: :: XPRIMET1:201

for p being Prime holds
( not p < 557 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 )

proof end;

theorem :: XPRIMET1:202

for k being Nat
for p being Prime st p * p <= k & k < 310249 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 holds
p = 547

proof end;

theorem Ttool563a: :: XPRIMET1:203

for p being Prime holds
( not p < 563 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 )

proof end;

theorem :: XPRIMET1:204

for k being Nat
for p being Prime st p * p <= k & k < 316969 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 holds
p = 557

proof end;

theorem Ttool569a: :: XPRIMET1:205

for p being Prime holds
( not p < 569 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 )

proof end;

theorem :: XPRIMET1:206

for k being Nat
for p being Prime st p * p <= k & k < 323761 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 holds
p = 563

proof end;

theorem Ttool571a: :: XPRIMET1:207

for p being Prime holds
( not p < 571 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 )

proof end;

theorem :: XPRIMET1:208

for k being Nat
for p being Prime st p * p <= k & k < 326041 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 holds
p = 569

proof end;

theorem Ttool577a: :: XPRIMET1:209

for p being Prime holds
( not p < 577 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 )

proof end;

theorem :: XPRIMET1:210

for k being Nat
for p being Prime st p * p <= k & k < 332929 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 holds
p = 571

proof end;

theorem Ttool587a: :: XPRIMET1:211

for p being Prime holds
( not p < 587 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 )

proof end;

theorem :: XPRIMET1:212

for k being Nat
for p being Prime st p * p <= k & k < 344569 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 holds
p = 577

proof end;

theorem Ttool593a: :: XPRIMET1:213

for p being Prime holds
( not p < 593 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 )

proof end;

theorem :: XPRIMET1:214

for k being Nat
for p being Prime st p * p <= k & k < 351649 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 holds
p = 587

proof end;

theorem Ttool599a: :: XPRIMET1:215

for p being Prime holds
( not p < 599 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 )

proof end;

theorem :: XPRIMET1:216

for k being Nat
for p being Prime st p * p <= k & k < 358801 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 holds
p = 593

proof end;

theorem Ttool601a: :: XPRIMET1:217

for p being Prime holds
( not p < 601 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 )

proof end;

theorem :: XPRIMET1:218

for k being Nat
for p being Prime st p * p <= k & k < 361201 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 holds
p = 599

proof end;

theorem Ttool607a: :: XPRIMET1:219

for p being Prime holds
( not p < 607 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 )

proof end;

theorem :: XPRIMET1:220

for k being Nat
for p being Prime st p * p <= k & k < 368449 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 holds
p = 601

proof end;

theorem Ttool613a: :: XPRIMET1:221

for p being Prime holds
( not p < 613 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 )

proof end;

theorem :: XPRIMET1:222

for k being Nat
for p being Prime st p * p <= k & k < 375769 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 holds
p = 607

proof end;

theorem Ttool617a: :: XPRIMET1:223

for p being Prime holds
( not p < 617 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 )

proof end;

theorem :: XPRIMET1:224

for k being Nat
for p being Prime st p * p <= k & k < 380689 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 holds
p = 613

proof end;

theorem Ttool619a: :: XPRIMET1:225

for p being Prime holds
( not p < 619 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 )

proof end;

theorem :: XPRIMET1:226

for k being Nat
for p being Prime st p * p <= k & k < 383161 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 holds
p = 617

proof end;

theorem Ttool631a: :: XPRIMET1:227

for p being Prime holds
( not p < 631 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 )

proof end;

theorem :: XPRIMET1:228

for k being Nat
for p being Prime st p * p <= k & k < 398161 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 holds
p = 619

proof end;

theorem Ttool641a: :: XPRIMET1:229

for p being Prime holds
( not p < 641 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 )

proof end;

theorem :: XPRIMET1:230

for k being Nat
for p being Prime st p * p <= k & k < 410881 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 holds
p = 631

proof end;

theorem Ttool643a: :: XPRIMET1:231

for p being Prime holds
( not p < 643 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 )

proof end;

theorem :: XPRIMET1:232

for k being Nat
for p being Prime st p * p <= k & k < 413449 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 holds
p = 641

proof end;

theorem Ttool647a: :: XPRIMET1:233

for p being Prime holds
( not p < 647 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 )

proof end;

theorem :: XPRIMET1:234

for k being Nat
for p being Prime st p * p <= k & k < 418609 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 holds
p = 643

proof end;

theorem Ttool653a: :: XPRIMET1:235

for p being Prime holds
( not p < 653 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 )

proof end;

theorem :: XPRIMET1:236

for k being Nat
for p being Prime st p * p <= k & k < 426409 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 holds
p = 647

proof end;

theorem Ttool659a: :: XPRIMET1:237

for p being Prime holds
( not p < 659 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 )

proof end;

theorem :: XPRIMET1:238

for k being Nat
for p being Prime st p * p <= k & k < 434281 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 holds
p = 653

proof end;

theorem Ttool661a: :: XPRIMET1:239

for p being Prime holds
( not p < 661 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 )

proof end;

theorem :: XPRIMET1:240

for k being Nat
for p being Prime st p * p <= k & k < 436921 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 holds
p = 659

proof end;

theorem Ttool673a: :: XPRIMET1:241

for p being Prime holds
( not p < 673 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 )

proof end;

theorem :: XPRIMET1:242

for k being Nat
for p being Prime st p * p <= k & k < 452929 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 holds
p = 661

proof end;

theorem Ttool677a: :: XPRIMET1:243

for p being Prime holds
( not p < 677 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 )

proof end;

theorem :: XPRIMET1:244

for k being Nat
for p being Prime st p * p <= k & k < 458329 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 holds
p = 673

proof end;

theorem Ttool683a: :: XPRIMET1:245

for p being Prime holds
( not p < 683 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 )

proof end;

theorem :: XPRIMET1:246

for k being Nat
for p being Prime st p * p <= k & k < 466489 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 holds
p = 677

proof end;

theorem Ttool691a: :: XPRIMET1:247

for p being Prime holds
( not p < 691 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 )

proof end;

theorem :: XPRIMET1:248

for k being Nat
for p being Prime st p * p <= k & k < 477481 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 holds
p = 683

proof end;

theorem Ttool701a: :: XPRIMET1:249

for p being Prime holds
( not p < 701 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 )

proof end;

theorem :: XPRIMET1:250

for k being Nat
for p being Prime st p * p <= k & k < 491401 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 holds
p = 691

proof end;

theorem Ttool709a: :: XPRIMET1:251

for p being Prime holds
( not p < 709 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 )

proof end;

theorem :: XPRIMET1:252

for k being Nat
for p being Prime st p * p <= k & k < 502681 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 holds
p = 701

proof end;

theorem Ttool719a: :: XPRIMET1:253

for p being Prime holds
( not p < 719 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 )

proof end;

theorem :: XPRIMET1:254

for k being Nat
for p being Prime st p * p <= k & k < 516961 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 holds
p = 709

proof end;

theorem Ttool727a: :: XPRIMET1:255

for p being Prime holds
( not p < 727 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 )

proof end;

theorem :: XPRIMET1:256

for k being Nat
for p being Prime st p * p <= k & k < 528529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 holds
p = 719

proof end;

theorem Ttool733a: :: XPRIMET1:257

for p being Prime holds
( not p < 733 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 )

proof end;

theorem :: XPRIMET1:258

for k being Nat
for p being Prime st p * p <= k & k < 537289 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 holds
p = 727

proof end;

theorem Ttool739a: :: XPRIMET1:259

for p being Prime holds
( not p < 739 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 )

proof end;

theorem :: XPRIMET1:260

for k being Nat
for p being Prime st p * p <= k & k < 546121 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 holds
p = 733

proof end;

theorem Ttool743a: :: XPRIMET1:261

for p being Prime holds
( not p < 743 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 )

proof end;

theorem :: XPRIMET1:262

for k being Nat
for p being Prime st p * p <= k & k < 552049 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 holds
p = 739

proof end;

theorem Ttool751a: :: XPRIMET1:263

for p being Prime holds
( not p < 751 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 )

proof end;

theorem :: XPRIMET1:264

for k being Nat
for p being Prime st p * p <= k & k < 564001 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 holds
p = 743

proof end;

theorem Ttool757a: :: XPRIMET1:265

for p being Prime holds
( not p < 757 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 )

proof end;

theorem :: XPRIMET1:266

for k being Nat
for p being Prime st p * p <= k & k < 573049 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 holds
p = 751

proof end;

theorem Ttool761a: :: XPRIMET1:267

for p being Prime holds
( not p < 761 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 )

proof end;

theorem :: XPRIMET1:268

for k being Nat
for p being Prime st p * p <= k & k < 579121 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 holds
p = 757

proof end;

theorem Ttool769a: :: XPRIMET1:269

for p being Prime holds
( not p < 769 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 )

proof end;

theorem :: XPRIMET1:270

for k being Nat
for p being Prime st p * p <= k & k < 591361 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 holds
p = 761

proof end;

theorem Ttool773a: :: XPRIMET1:271

for p being Prime holds
( not p < 773 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 )

proof end;

theorem :: XPRIMET1:272

for k being Nat
for p being Prime st p * p <= k & k < 597529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 holds
p = 769

proof end;

theorem Ttool787a: :: XPRIMET1:273

for p being Prime holds
( not p < 787 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 )

proof end;

theorem :: XPRIMET1:274

for k being Nat
for p being Prime st p * p <= k & k < 619369 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 holds
p = 773

proof end;

theorem Ttool797a: :: XPRIMET1:275

for p being Prime holds
( not p < 797 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 )

proof end;

theorem :: XPRIMET1:276

for k being Nat
for p being Prime st p * p <= k & k < 635209 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 holds
p = 787

proof end;

theorem Ttool809a: :: XPRIMET1:277

for p being Prime holds
( not p < 809 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 )

proof end;

theorem :: XPRIMET1:278

for k being Nat
for p being Prime st p * p <= k & k < 654481 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 holds
p = 797

proof end;

theorem Ttool811a: :: XPRIMET1:279

for p being Prime holds
( not p < 811 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 )

proof end;

theorem :: XPRIMET1:280

for k being Nat
for p being Prime st p * p <= k & k < 657721 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 holds
p = 809

proof end;

theorem Ttool821a: :: XPRIMET1:281

for p being Prime holds
( not p < 821 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 )

proof end;

theorem :: XPRIMET1:282

for k being Nat
for p being Prime st p * p <= k & k < 674041 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 holds
p = 811

proof end;

theorem Ttool823a: :: XPRIMET1:283

for p being Prime holds
( not p < 823 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 )

proof end;

theorem :: XPRIMET1:284

for k being Nat
for p being Prime st p * p <= k & k < 677329 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 holds
p = 821

proof end;

theorem Ttool827a: :: XPRIMET1:285

for p being Prime holds
( not p < 827 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 )

proof end;

theorem :: XPRIMET1:286

for k being Nat
for p being Prime st p * p <= k & k < 683929 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 holds
p = 823

proof end;

theorem Ttool829a: :: XPRIMET1:287

for p being Prime holds
( not p < 829 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 )

proof end;

theorem :: XPRIMET1:288

for k being Nat
for p being Prime st p * p <= k & k < 687241 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 holds
p = 827

proof end;

theorem Ttool839a: :: XPRIMET1:289

for p being Prime holds
( not p < 839 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 )

proof end;

theorem :: XPRIMET1:290

for k being Nat
for p being Prime st p * p <= k & k < 703921 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 holds
p = 829

proof end;

theorem Ttool853a: :: XPRIMET1:291

for p being Prime holds
( not p < 853 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 )

proof end;

theorem :: XPRIMET1:292

for k being Nat
for p being Prime st p * p <= k & k < 727609 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 holds
p = 839

proof end;

theorem Ttool857a: :: XPRIMET1:293

for p being Prime holds
( not p < 857 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 )

proof end;

theorem :: XPRIMET1:294

for k being Nat
for p being Prime st p * p <= k & k < 734449 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 holds
p = 853

proof end;

theorem Ttool859a: :: XPRIMET1:295

for p being Prime holds
( not p < 859 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 )

proof end;

theorem :: XPRIMET1:296

for k being Nat
for p being Prime st p * p <= k & k < 737881 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 holds
p = 857

proof end;

theorem Ttool863a: :: XPRIMET1:297

for p being Prime holds
( not p < 863 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 )

proof end;

theorem :: XPRIMET1:298

for k being Nat
for p being Prime st p * p <= k & k < 744769 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 holds
p = 859

proof end;

theorem Ttool877a: :: XPRIMET1:299

for p being Prime holds
( not p < 877 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 )

proof end;

theorem :: XPRIMET1:300

for k being Nat
for p being Prime st p * p <= k & k < 769129 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 holds
p = 863

proof end;

theorem Ttool881a: :: XPRIMET1:301

for p being Prime holds
( not p < 881 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 )

proof end;

theorem :: XPRIMET1:302

for k being Nat
for p being Prime st p * p <= k & k < 776161 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 holds
p = 877

proof end;

theorem Ttool883a: :: XPRIMET1:303

for p being Prime holds
( not p < 883 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 )

proof end;

theorem :: XPRIMET1:304

for k being Nat
for p being Prime st p * p <= k & k < 779689 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 holds
p = 881

proof end;

theorem Ttool887a: :: XPRIMET1:305

for p being Prime holds
( not p < 887 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 )

proof end;

theorem :: XPRIMET1:306

for k being Nat
for p being Prime st p * p <= k & k < 786769 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 holds
p = 883

proof end;

theorem Ttool907a: :: XPRIMET1:307

for p being Prime holds
( not p < 907 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 )

proof end;

theorem :: XPRIMET1:308

for k being Nat
for p being Prime st p * p <= k & k < 822649 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 holds
p = 887

proof end;

theorem Ttool911a: :: XPRIMET1:309

for p being Prime holds
( not p < 911 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 )

proof end;

theorem :: XPRIMET1:310

for k being Nat
for p being Prime st p * p <= k & k < 829921 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 holds
p = 907

proof end;

theorem Ttool919a: :: XPRIMET1:311

for p being Prime holds
( not p < 919 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 )

proof end;

theorem :: XPRIMET1:312

for k being Nat
for p being Prime st p * p <= k & k < 844561 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 holds
p = 911

proof end;

theorem Ttool929a: :: XPRIMET1:313

for p being Prime holds
( not p < 929 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 )

proof end;

theorem :: XPRIMET1:314

for k being Nat
for p being Prime st p * p <= k & k < 863041 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 holds
p = 919

proof end;

theorem Ttool937a: :: XPRIMET1:315

for p being Prime holds
( not p < 937 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 )

proof end;

theorem :: XPRIMET1:316

for k being Nat
for p being Prime st p * p <= k & k < 877969 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 holds
p = 929

proof end;

theorem Ttool941a: :: XPRIMET1:317

for p being Prime holds
( not p < 941 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 )

proof end;

theorem :: XPRIMET1:318

for k being Nat
for p being Prime st p * p <= k & k < 885481 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 holds
p = 937

proof end;

theorem Ttool947a: :: XPRIMET1:319

for p being Prime holds
( not p < 947 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 )

proof end;

theorem :: XPRIMET1:320

for k being Nat
for p being Prime st p * p <= k & k < 896809 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 holds
p = 941

proof end;

theorem Ttool953a: :: XPRIMET1:321

for p being Prime holds
( not p < 953 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 )

proof end;

theorem :: XPRIMET1:322

for k being Nat
for p being Prime st p * p <= k & k < 908209 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 holds
p = 947

proof end;

theorem Ttool967a: :: XPRIMET1:323

for p being Prime holds
( not p < 967 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 )

proof end;

theorem :: XPRIMET1:324

for k being Nat
for p being Prime st p * p <= k & k < 935089 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 holds
p = 953

proof end;

theorem Ttool971a: :: XPRIMET1:325

for p being Prime holds
( not p < 971 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 )

proof end;

theorem :: XPRIMET1:326

for k being Nat
for p being Prime st p * p <= k & k < 942841 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 holds
p = 967

proof end;

theorem Ttool977a: :: XPRIMET1:327

for p being Prime holds
( not p < 977 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 or p = 971 )

proof end;

theorem :: XPRIMET1:328

for k being Nat
for p being Prime st p * p <= k & k < 954529 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 & not p = 967 holds
p = 971

proof end;

theorem Ttool983a: :: XPRIMET1:329

for p being Prime holds
( not p < 983 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 or p = 971 or p = 977 )

proof end;

theorem :: XPRIMET1:330

for k being Nat
for p being Prime st p * p <= k & k < 966289 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 & not p = 967 & not p = 971 holds
p = 977

proof end;

theorem Ttool991a: :: XPRIMET1:331

for p being Prime holds
( not p < 991 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 or p = 971 or p = 977 or p = 983 )

proof end;

theorem :: XPRIMET1:332

for k being Nat
for p being Prime st p * p <= k & k < 982081 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 & not p = 967 & not p = 971 & not p = 977 holds
p = 983

proof end;

theorem Ttool997a: :: XPRIMET1:333

for p being Prime holds
( not p < 997 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991 )

proof end;

theorem :: XPRIMET1:334

for k being Nat
for p being Prime st p * p <= k & k < 994009 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 & not p = 967 & not p = 971 & not p = 977 & not p = 983 holds
p = 991

proof end;

theorem Ttool1009a: :: XPRIMET1:335

for p being Prime holds
( not p < 1009 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or p = 269 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or p = 311 or p = 313 or p = 317 or p = 331 or p = 337 or p = 347 or p = 349 or p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or p = 523 or p = 541 or p = 547 or p = 557 or p = 563 or p = 569 or p = 571 or p = 577 or p = 587 or p = 593 or p = 599 or p = 601 or p = 607 or p = 613 or p = 617 or p = 619 or p = 631 or p = 641 or p = 643 or p = 647 or p = 653 or p = 659 or p = 661 or p = 673 or p = 677 or p = 683 or p = 691 or p = 701 or p = 709 or p = 719 or p = 727 or p = 733 or p = 739 or p = 743 or p = 751 or p = 757 or p = 761 or p = 769 or p = 773 or p = 787 or p = 797 or p = 809 or p = 811 or p = 821 or p = 823 or p = 827 or p = 829 or p = 839 or p = 853 or p = 857 or p = 859 or p = 863 or p = 877 or p = 881 or p = 883 or p = 887 or p = 907 or p = 911 or p = 919 or p = 929 or p = 937 or p = 941 or p = 947 or p = 953 or p = 967 or p = 971 or p = 977 or p = 983 or p = 991 or p = 997 )

proof end;

theorem :: XPRIMET1:336

for k being Nat
for p being Prime st p * p <= k & k < 1018081 & not p = 2 & not p = 3 & not p = 5 & not p = 7 & not p = 11 & not p = 13 & not p = 17 & not p = 19 & not p = 23 & not p = 29 & not p = 31 & not p = 37 & not p = 41 & not p = 43 & not p = 47 & not p = 53 & not p = 59 & not p = 61 & not p = 67 & not p = 71 & not p = 73 & not p = 79 & not p = 83 & not p = 89 & not p = 97 & not p = 101 & not p = 103 & not p = 107 & not p = 109 & not p = 113 & not p = 127 & not p = 131 & not p = 137 & not p = 139 & not p = 149 & not p = 151 & not p = 157 & not p = 163 & not p = 167 & not p = 173 & not p = 179 & not p = 181 & not p = 191 & not p = 193 & not p = 197 & not p = 199 & not p = 211 & not p = 223 & not p = 227 & not p = 229 & not p = 233 & not p = 239 & not p = 241 & not p = 251 & not p = 257 & not p = 263 & not p = 269 & not p = 271 & not p = 277 & not p = 281 & not p = 283 & not p = 293 & not p = 307 & not p = 311 & not p = 313 & not p = 317 & not p = 331 & not p = 337 & not p = 347 & not p = 349 & not p = 353 & not p = 359 & not p = 367 & not p = 373 & not p = 379 & not p = 383 & not p = 389 & not p = 397 & not p = 401 & not p = 409 & not p = 419 & not p = 421 & not p = 431 & not p = 433 & not p = 439 & not p = 443 & not p = 449 & not p = 457 & not p = 461 & not p = 463 & not p = 467 & not p = 479 & not p = 487 & not p = 491 & not p = 499 & not p = 503 & not p = 509 & not p = 521 & not p = 523 & not p = 541 & not p = 547 & not p = 557 & not p = 563 & not p = 569 & not p = 571 & not p = 577 & not p = 587 & not p = 593 & not p = 599 & not p = 601 & not p = 607 & not p = 613 & not p = 617 & not p = 619 & not p = 631 & not p = 641 & not p = 643 & not p = 647 & not p = 653 & not p = 659 & not p = 661 & not p = 673 & not p = 677 & not p = 683 & not p = 691 & not p = 701 & not p = 709 & not p = 719 & not p = 727 & not p = 733 & not p = 739 & not p = 743 & not p = 751 & not p = 757 & not p = 761 & not p = 769 & not p = 773 & not p = 787 & not p = 797 & not p = 809 & not p = 811 & not p = 821 & not p = 823 & not p = 827 & not p = 829 & not p = 839 & not p = 853 & not p = 857 & not p = 859 & not p = 863 & not p = 877 & not p = 881 & not p = 883 & not p = 887 & not p = 907 & not p = 911 & not p = 919 & not p = 929 & not p = 937 & not p = 941 & not p = 947 & not p = 953 & not p = 967 & not p = 971 & not p = 977 & not p = 983 & not p = 991 holds
p = 997

proof end;