
theorem
  8377 is prime
proof
  now
    8377 = 2*4188 + 1; hence not 2 divides 8377 by NAT_4:9;
    8377 = 3*2792 + 1; hence not 3 divides 8377 by NAT_4:9;
    8377 = 5*1675 + 2; hence not 5 divides 8377 by NAT_4:9;
    8377 = 7*1196 + 5; hence not 7 divides 8377 by NAT_4:9;
    8377 = 11*761 + 6; hence not 11 divides 8377 by NAT_4:9;
    8377 = 13*644 + 5; hence not 13 divides 8377 by NAT_4:9;
    8377 = 17*492 + 13; hence not 17 divides 8377 by NAT_4:9;
    8377 = 19*440 + 17; hence not 19 divides 8377 by NAT_4:9;
    8377 = 23*364 + 5; hence not 23 divides 8377 by NAT_4:9;
    8377 = 29*288 + 25; hence not 29 divides 8377 by NAT_4:9;
    8377 = 31*270 + 7; hence not 31 divides 8377 by NAT_4:9;
    8377 = 37*226 + 15; hence not 37 divides 8377 by NAT_4:9;
    8377 = 41*204 + 13; hence not 41 divides 8377 by NAT_4:9;
    8377 = 43*194 + 35; hence not 43 divides 8377 by NAT_4:9;
    8377 = 47*178 + 11; hence not 47 divides 8377 by NAT_4:9;
    8377 = 53*158 + 3; hence not 53 divides 8377 by NAT_4:9;
    8377 = 59*141 + 58; hence not 59 divides 8377 by NAT_4:9;
    8377 = 61*137 + 20; hence not 61 divides 8377 by NAT_4:9;
    8377 = 67*125 + 2; hence not 67 divides 8377 by NAT_4:9;
    8377 = 71*117 + 70; hence not 71 divides 8377 by NAT_4:9;
    8377 = 73*114 + 55; hence not 73 divides 8377 by NAT_4:9;
    8377 = 79*106 + 3; hence not 79 divides 8377 by NAT_4:9;
    8377 = 83*100 + 77; hence not 83 divides 8377 by NAT_4:9;
    8377 = 89*94 + 11; hence not 89 divides 8377 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8377 & n is prime
  holds not n divides 8377 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
