
theorem
  8179 is prime
proof
  now
    8179 = 2*4089 + 1; hence not 2 divides 8179 by NAT_4:9;
    8179 = 3*2726 + 1; hence not 3 divides 8179 by NAT_4:9;
    8179 = 5*1635 + 4; hence not 5 divides 8179 by NAT_4:9;
    8179 = 7*1168 + 3; hence not 7 divides 8179 by NAT_4:9;
    8179 = 11*743 + 6; hence not 11 divides 8179 by NAT_4:9;
    8179 = 13*629 + 2; hence not 13 divides 8179 by NAT_4:9;
    8179 = 17*481 + 2; hence not 17 divides 8179 by NAT_4:9;
    8179 = 19*430 + 9; hence not 19 divides 8179 by NAT_4:9;
    8179 = 23*355 + 14; hence not 23 divides 8179 by NAT_4:9;
    8179 = 29*282 + 1; hence not 29 divides 8179 by NAT_4:9;
    8179 = 31*263 + 26; hence not 31 divides 8179 by NAT_4:9;
    8179 = 37*221 + 2; hence not 37 divides 8179 by NAT_4:9;
    8179 = 41*199 + 20; hence not 41 divides 8179 by NAT_4:9;
    8179 = 43*190 + 9; hence not 43 divides 8179 by NAT_4:9;
    8179 = 47*174 + 1; hence not 47 divides 8179 by NAT_4:9;
    8179 = 53*154 + 17; hence not 53 divides 8179 by NAT_4:9;
    8179 = 59*138 + 37; hence not 59 divides 8179 by NAT_4:9;
    8179 = 61*134 + 5; hence not 61 divides 8179 by NAT_4:9;
    8179 = 67*122 + 5; hence not 67 divides 8179 by NAT_4:9;
    8179 = 71*115 + 14; hence not 71 divides 8179 by NAT_4:9;
    8179 = 73*112 + 3; hence not 73 divides 8179 by NAT_4:9;
    8179 = 79*103 + 42; hence not 79 divides 8179 by NAT_4:9;
    8179 = 83*98 + 45; hence not 83 divides 8179 by NAT_4:9;
    8179 = 89*91 + 80; hence not 89 divides 8179 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8179 & n is prime
  holds not n divides 8179 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
