2927 = (2 * 1463) + 1 ;
hence not 2 divides 2927 by NAT_4:9; :: thesis:
2927 = (3 * 975) + 2 ;
hence not 3 divides 2927 by NAT_4:9; :: thesis:
2927 = (5 * 585) + 2 ;
hence not 5 divides 2927 by NAT_4:9; :: thesis:
2927 = (7 * 418) + 1 ;
hence not 7 divides 2927 by NAT_4:9; :: thesis:
2927 = (11 * 266) + 1 ;
hence not 11 divides 2927 by NAT_4:9; :: thesis:
2927 = (13 * 225) + 2 ;
hence not 13 divides 2927 by NAT_4:9; :: thesis:
2927 = (17 * 172) + 3 ;
hence not 17 divides 2927 by NAT_4:9; :: thesis:
2927 = (19 * 154) + 1 ;
hence not 19 divides 2927 by NAT_4:9; :: thesis:
2927 = (23 * 127) + 6 ;
hence not 23 divides 2927 by NAT_4:9; :: thesis:
2927 = (29 * 100) + 27 ;
hence not 29 divides 2927 by NAT_4:9; :: thesis:
2927 = (31 * 94) + 13 ;
hence not 31 divides 2927 by NAT_4:9; :: thesis:
2927 = (37 * 79) + 4 ;
hence not 37 divides 2927 by NAT_4:9; :: thesis:
2927 = (41 * 71) + 16 ;
hence not 41 divides 2927 by NAT_4:9; :: thesis:
2927 = (43 * 68) + 3 ;
hence not 43 divides 2927 by NAT_4:9; :: thesis:
2927 = (47 * 62) + 13 ;
hence not 47 divides 2927 by NAT_4:9; :: thesis:
2927 = (53 * 55) + 12 ;
hence not 53 divides 2927 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2927 & n is prime holds
not n divides 2927 by XPRIMET1:32;
hence 2927 is prime by NAT_4:14; :: thesis: