
theorem
  3109 is prime
proof
  now
    3109 = 2*1554 + 1; hence not 2 divides 3109 by NAT_4:9;
    3109 = 3*1036 + 1; hence not 3 divides 3109 by NAT_4:9;
    3109 = 5*621 + 4; hence not 5 divides 3109 by NAT_4:9;
    3109 = 7*444 + 1; hence not 7 divides 3109 by NAT_4:9;
    3109 = 11*282 + 7; hence not 11 divides 3109 by NAT_4:9;
    3109 = 13*239 + 2; hence not 13 divides 3109 by NAT_4:9;
    3109 = 17*182 + 15; hence not 17 divides 3109 by NAT_4:9;
    3109 = 19*163 + 12; hence not 19 divides 3109 by NAT_4:9;
    3109 = 23*135 + 4; hence not 23 divides 3109 by NAT_4:9;
    3109 = 29*107 + 6; hence not 29 divides 3109 by NAT_4:9;
    3109 = 31*100 + 9; hence not 31 divides 3109 by NAT_4:9;
    3109 = 37*84 + 1; hence not 37 divides 3109 by NAT_4:9;
    3109 = 41*75 + 34; hence not 41 divides 3109 by NAT_4:9;
    3109 = 43*72 + 13; hence not 43 divides 3109 by NAT_4:9;
    3109 = 47*66 + 7; hence not 47 divides 3109 by NAT_4:9;
    3109 = 53*58 + 35; hence not 53 divides 3109 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 3109 & n is prime
  holds not n divides 3109 by XPRIMET1:32;
  hence thesis by NAT_4:14;
end;
