
theorem
  8369 is prime
proof
  now
    8369 = 2*4184 + 1; hence not 2 divides 8369 by NAT_4:9;
    8369 = 3*2789 + 2; hence not 3 divides 8369 by NAT_4:9;
    8369 = 5*1673 + 4; hence not 5 divides 8369 by NAT_4:9;
    8369 = 7*1195 + 4; hence not 7 divides 8369 by NAT_4:9;
    8369 = 11*760 + 9; hence not 11 divides 8369 by NAT_4:9;
    8369 = 13*643 + 10; hence not 13 divides 8369 by NAT_4:9;
    8369 = 17*492 + 5; hence not 17 divides 8369 by NAT_4:9;
    8369 = 19*440 + 9; hence not 19 divides 8369 by NAT_4:9;
    8369 = 23*363 + 20; hence not 23 divides 8369 by NAT_4:9;
    8369 = 29*288 + 17; hence not 29 divides 8369 by NAT_4:9;
    8369 = 31*269 + 30; hence not 31 divides 8369 by NAT_4:9;
    8369 = 37*226 + 7; hence not 37 divides 8369 by NAT_4:9;
    8369 = 41*204 + 5; hence not 41 divides 8369 by NAT_4:9;
    8369 = 43*194 + 27; hence not 43 divides 8369 by NAT_4:9;
    8369 = 47*178 + 3; hence not 47 divides 8369 by NAT_4:9;
    8369 = 53*157 + 48; hence not 53 divides 8369 by NAT_4:9;
    8369 = 59*141 + 50; hence not 59 divides 8369 by NAT_4:9;
    8369 = 61*137 + 12; hence not 61 divides 8369 by NAT_4:9;
    8369 = 67*124 + 61; hence not 67 divides 8369 by NAT_4:9;
    8369 = 71*117 + 62; hence not 71 divides 8369 by NAT_4:9;
    8369 = 73*114 + 47; hence not 73 divides 8369 by NAT_4:9;
    8369 = 79*105 + 74; hence not 79 divides 8369 by NAT_4:9;
    8369 = 83*100 + 69; hence not 83 divides 8369 by NAT_4:9;
    8369 = 89*94 + 3; hence not 89 divides 8369 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8369 & n is prime
  holds not n divides 8369 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
