
theorem
  8317 is prime
proof
  now
    8317 = 2*4158 + 1; hence not 2 divides 8317 by NAT_4:9;
    8317 = 3*2772 + 1; hence not 3 divides 8317 by NAT_4:9;
    8317 = 5*1663 + 2; hence not 5 divides 8317 by NAT_4:9;
    8317 = 7*1188 + 1; hence not 7 divides 8317 by NAT_4:9;
    8317 = 11*756 + 1; hence not 11 divides 8317 by NAT_4:9;
    8317 = 13*639 + 10; hence not 13 divides 8317 by NAT_4:9;
    8317 = 17*489 + 4; hence not 17 divides 8317 by NAT_4:9;
    8317 = 19*437 + 14; hence not 19 divides 8317 by NAT_4:9;
    8317 = 23*361 + 14; hence not 23 divides 8317 by NAT_4:9;
    8317 = 29*286 + 23; hence not 29 divides 8317 by NAT_4:9;
    8317 = 31*268 + 9; hence not 31 divides 8317 by NAT_4:9;
    8317 = 37*224 + 29; hence not 37 divides 8317 by NAT_4:9;
    8317 = 41*202 + 35; hence not 41 divides 8317 by NAT_4:9;
    8317 = 43*193 + 18; hence not 43 divides 8317 by NAT_4:9;
    8317 = 47*176 + 45; hence not 47 divides 8317 by NAT_4:9;
    8317 = 53*156 + 49; hence not 53 divides 8317 by NAT_4:9;
    8317 = 59*140 + 57; hence not 59 divides 8317 by NAT_4:9;
    8317 = 61*136 + 21; hence not 61 divides 8317 by NAT_4:9;
    8317 = 67*124 + 9; hence not 67 divides 8317 by NAT_4:9;
    8317 = 71*117 + 10; hence not 71 divides 8317 by NAT_4:9;
    8317 = 73*113 + 68; hence not 73 divides 8317 by NAT_4:9;
    8317 = 79*105 + 22; hence not 79 divides 8317 by NAT_4:9;
    8317 = 83*100 + 17; hence not 83 divides 8317 by NAT_4:9;
    8317 = 89*93 + 40; hence not 89 divides 8317 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8317 & n is prime
  holds not n divides 8317 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
