2917 = (2 * 1458) + 1 ;
hence not 2 divides 2917 by NAT_4:9; :: thesis:
2917 = (3 * 972) + 1 ;
hence not 3 divides 2917 by NAT_4:9; :: thesis:
2917 = (5 * 583) + 2 ;
hence not 5 divides 2917 by NAT_4:9; :: thesis:
2917 = (7 * 416) + 5 ;
hence not 7 divides 2917 by NAT_4:9; :: thesis:
2917 = (11 * 265) + 2 ;
hence not 11 divides 2917 by NAT_4:9; :: thesis:
2917 = (13 * 224) + 5 ;
hence not 13 divides 2917 by NAT_4:9; :: thesis:
2917 = (17 * 171) + 10 ;
hence not 17 divides 2917 by NAT_4:9; :: thesis:
2917 = (19 * 153) + 10 ;
hence not 19 divides 2917 by NAT_4:9; :: thesis:
2917 = (23 * 126) + 19 ;
hence not 23 divides 2917 by NAT_4:9; :: thesis:
2917 = (29 * 100) + 17 ;
hence not 29 divides 2917 by NAT_4:9; :: thesis:
2917 = (31 * 94) + 3 ;
hence not 31 divides 2917 by NAT_4:9; :: thesis:
2917 = (37 * 78) + 31 ;
hence not 37 divides 2917 by NAT_4:9; :: thesis:
2917 = (41 * 71) + 6 ;
hence not 41 divides 2917 by NAT_4:9; :: thesis:
2917 = (43 * 67) + 36 ;
hence not 43 divides 2917 by NAT_4:9; :: thesis:
2917 = (47 * 62) + 3 ;
hence not 47 divides 2917 by NAT_4:9; :: thesis:
2917 = (53 * 55) + 2 ;
hence not 53 divides 2917 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2917 & n is prime holds
not n divides 2917 by XPRIMET1:32;
hence 2917 is prime by NAT_4:14; :: thesis: