
theorem
  8363 is prime
proof
  now
    8363 = 2*4181 + 1; hence not 2 divides 8363 by NAT_4:9;
    8363 = 3*2787 + 2; hence not 3 divides 8363 by NAT_4:9;
    8363 = 5*1672 + 3; hence not 5 divides 8363 by NAT_4:9;
    8363 = 7*1194 + 5; hence not 7 divides 8363 by NAT_4:9;
    8363 = 11*760 + 3; hence not 11 divides 8363 by NAT_4:9;
    8363 = 13*643 + 4; hence not 13 divides 8363 by NAT_4:9;
    8363 = 17*491 + 16; hence not 17 divides 8363 by NAT_4:9;
    8363 = 19*440 + 3; hence not 19 divides 8363 by NAT_4:9;
    8363 = 23*363 + 14; hence not 23 divides 8363 by NAT_4:9;
    8363 = 29*288 + 11; hence not 29 divides 8363 by NAT_4:9;
    8363 = 31*269 + 24; hence not 31 divides 8363 by NAT_4:9;
    8363 = 37*226 + 1; hence not 37 divides 8363 by NAT_4:9;
    8363 = 41*203 + 40; hence not 41 divides 8363 by NAT_4:9;
    8363 = 43*194 + 21; hence not 43 divides 8363 by NAT_4:9;
    8363 = 47*177 + 44; hence not 47 divides 8363 by NAT_4:9;
    8363 = 53*157 + 42; hence not 53 divides 8363 by NAT_4:9;
    8363 = 59*141 + 44; hence not 59 divides 8363 by NAT_4:9;
    8363 = 61*137 + 6; hence not 61 divides 8363 by NAT_4:9;
    8363 = 67*124 + 55; hence not 67 divides 8363 by NAT_4:9;
    8363 = 71*117 + 56; hence not 71 divides 8363 by NAT_4:9;
    8363 = 73*114 + 41; hence not 73 divides 8363 by NAT_4:9;
    8363 = 79*105 + 68; hence not 79 divides 8363 by NAT_4:9;
    8363 = 83*100 + 63; hence not 83 divides 8363 by NAT_4:9;
    8363 = 89*93 + 86; hence not 89 divides 8363 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8363 & n is prime
  holds not n divides 8363 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
