
theorem
  8627 is prime
proof
  now
    8627 = 2*4313 + 1; hence not 2 divides 8627 by NAT_4:9;
    8627 = 3*2875 + 2; hence not 3 divides 8627 by NAT_4:9;
    8627 = 5*1725 + 2; hence not 5 divides 8627 by NAT_4:9;
    8627 = 7*1232 + 3; hence not 7 divides 8627 by NAT_4:9;
    8627 = 11*784 + 3; hence not 11 divides 8627 by NAT_4:9;
    8627 = 13*663 + 8; hence not 13 divides 8627 by NAT_4:9;
    8627 = 17*507 + 8; hence not 17 divides 8627 by NAT_4:9;
    8627 = 19*454 + 1; hence not 19 divides 8627 by NAT_4:9;
    8627 = 23*375 + 2; hence not 23 divides 8627 by NAT_4:9;
    8627 = 29*297 + 14; hence not 29 divides 8627 by NAT_4:9;
    8627 = 31*278 + 9; hence not 31 divides 8627 by NAT_4:9;
    8627 = 37*233 + 6; hence not 37 divides 8627 by NAT_4:9;
    8627 = 41*210 + 17; hence not 41 divides 8627 by NAT_4:9;
    8627 = 43*200 + 27; hence not 43 divides 8627 by NAT_4:9;
    8627 = 47*183 + 26; hence not 47 divides 8627 by NAT_4:9;
    8627 = 53*162 + 41; hence not 53 divides 8627 by NAT_4:9;
    8627 = 59*146 + 13; hence not 59 divides 8627 by NAT_4:9;
    8627 = 61*141 + 26; hence not 61 divides 8627 by NAT_4:9;
    8627 = 67*128 + 51; hence not 67 divides 8627 by NAT_4:9;
    8627 = 71*121 + 36; hence not 71 divides 8627 by NAT_4:9;
    8627 = 73*118 + 13; hence not 73 divides 8627 by NAT_4:9;
    8627 = 79*109 + 16; hence not 79 divides 8627 by NAT_4:9;
    8627 = 83*103 + 78; hence not 83 divides 8627 by NAT_4:9;
    8627 = 89*96 + 83; hence not 89 divides 8627 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8627 & n is prime
  holds not n divides 8627 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
