
theorem
  8311 is prime
proof
  now
    8311 = 2*4155 + 1; hence not 2 divides 8311 by NAT_4:9;
    8311 = 3*2770 + 1; hence not 3 divides 8311 by NAT_4:9;
    8311 = 5*1662 + 1; hence not 5 divides 8311 by NAT_4:9;
    8311 = 7*1187 + 2; hence not 7 divides 8311 by NAT_4:9;
    8311 = 11*755 + 6; hence not 11 divides 8311 by NAT_4:9;
    8311 = 13*639 + 4; hence not 13 divides 8311 by NAT_4:9;
    8311 = 17*488 + 15; hence not 17 divides 8311 by NAT_4:9;
    8311 = 19*437 + 8; hence not 19 divides 8311 by NAT_4:9;
    8311 = 23*361 + 8; hence not 23 divides 8311 by NAT_4:9;
    8311 = 29*286 + 17; hence not 29 divides 8311 by NAT_4:9;
    8311 = 31*268 + 3; hence not 31 divides 8311 by NAT_4:9;
    8311 = 37*224 + 23; hence not 37 divides 8311 by NAT_4:9;
    8311 = 41*202 + 29; hence not 41 divides 8311 by NAT_4:9;
    8311 = 43*193 + 12; hence not 43 divides 8311 by NAT_4:9;
    8311 = 47*176 + 39; hence not 47 divides 8311 by NAT_4:9;
    8311 = 53*156 + 43; hence not 53 divides 8311 by NAT_4:9;
    8311 = 59*140 + 51; hence not 59 divides 8311 by NAT_4:9;
    8311 = 61*136 + 15; hence not 61 divides 8311 by NAT_4:9;
    8311 = 67*124 + 3; hence not 67 divides 8311 by NAT_4:9;
    8311 = 71*117 + 4; hence not 71 divides 8311 by NAT_4:9;
    8311 = 73*113 + 62; hence not 73 divides 8311 by NAT_4:9;
    8311 = 79*105 + 16; hence not 79 divides 8311 by NAT_4:9;
    8311 = 83*100 + 11; hence not 83 divides 8311 by NAT_4:9;
    8311 = 89*93 + 34; hence not 89 divides 8311 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8311 & n is prime
  holds not n divides 8311 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
