
theorem
  8677 is prime
proof
  now
    8677 = 2*4338 + 1; hence not 2 divides 8677 by NAT_4:9;
    8677 = 3*2892 + 1; hence not 3 divides 8677 by NAT_4:9;
    8677 = 5*1735 + 2; hence not 5 divides 8677 by NAT_4:9;
    8677 = 7*1239 + 4; hence not 7 divides 8677 by NAT_4:9;
    8677 = 11*788 + 9; hence not 11 divides 8677 by NAT_4:9;
    8677 = 13*667 + 6; hence not 13 divides 8677 by NAT_4:9;
    8677 = 17*510 + 7; hence not 17 divides 8677 by NAT_4:9;
    8677 = 19*456 + 13; hence not 19 divides 8677 by NAT_4:9;
    8677 = 23*377 + 6; hence not 23 divides 8677 by NAT_4:9;
    8677 = 29*299 + 6; hence not 29 divides 8677 by NAT_4:9;
    8677 = 31*279 + 28; hence not 31 divides 8677 by NAT_4:9;
    8677 = 37*234 + 19; hence not 37 divides 8677 by NAT_4:9;
    8677 = 41*211 + 26; hence not 41 divides 8677 by NAT_4:9;
    8677 = 43*201 + 34; hence not 43 divides 8677 by NAT_4:9;
    8677 = 47*184 + 29; hence not 47 divides 8677 by NAT_4:9;
    8677 = 53*163 + 38; hence not 53 divides 8677 by NAT_4:9;
    8677 = 59*147 + 4; hence not 59 divides 8677 by NAT_4:9;
    8677 = 61*142 + 15; hence not 61 divides 8677 by NAT_4:9;
    8677 = 67*129 + 34; hence not 67 divides 8677 by NAT_4:9;
    8677 = 71*122 + 15; hence not 71 divides 8677 by NAT_4:9;
    8677 = 73*118 + 63; hence not 73 divides 8677 by NAT_4:9;
    8677 = 79*109 + 66; hence not 79 divides 8677 by NAT_4:9;
    8677 = 83*104 + 45; hence not 83 divides 8677 by NAT_4:9;
    8677 = 89*97 + 44; hence not 89 divides 8677 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8677 & n is prime
  holds not n divides 8677 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
