2909 = (2 * 1454) + 1 ;
hence not 2 divides 2909 by NAT_4:9; :: thesis:
2909 = (3 * 969) + 2 ;
hence not 3 divides 2909 by NAT_4:9; :: thesis:
2909 = (5 * 581) + 4 ;
hence not 5 divides 2909 by NAT_4:9; :: thesis:
2909 = (7 * 415) + 4 ;
hence not 7 divides 2909 by NAT_4:9; :: thesis:
2909 = (11 * 264) + 5 ;
hence not 11 divides 2909 by NAT_4:9; :: thesis:
2909 = (13 * 223) + 10 ;
hence not 13 divides 2909 by NAT_4:9; :: thesis:
2909 = (17 * 171) + 2 ;
hence not 17 divides 2909 by NAT_4:9; :: thesis:
2909 = (19 * 153) + 2 ;
hence not 19 divides 2909 by NAT_4:9; :: thesis:
2909 = (23 * 126) + 11 ;
hence not 23 divides 2909 by NAT_4:9; :: thesis:
2909 = (29 * 100) + 9 ;
hence not 29 divides 2909 by NAT_4:9; :: thesis:
2909 = (31 * 93) + 26 ;
hence not 31 divides 2909 by NAT_4:9; :: thesis:
2909 = (37 * 78) + 23 ;
hence not 37 divides 2909 by NAT_4:9; :: thesis:
2909 = (41 * 70) + 39 ;
hence not 41 divides 2909 by NAT_4:9; :: thesis:
2909 = (43 * 67) + 28 ;
hence not 43 divides 2909 by NAT_4:9; :: thesis:
2909 = (47 * 61) + 42 ;
hence not 47 divides 2909 by NAT_4:9; :: thesis:
2909 = (53 * 54) + 47 ;
hence not 53 divides 2909 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2909 & n is prime holds
not n divides 2909 by XPRIMET1:32;
hence 2909 is prime by NAT_4:14; :: thesis: