
theorem
  2609 is prime
proof
  now
    2609 = 2*1304 + 1; hence not 2 divides 2609 by NAT_4:9;
    2609 = 3*869 + 2; hence not 3 divides 2609 by NAT_4:9;
    2609 = 5*521 + 4; hence not 5 divides 2609 by NAT_4:9;
    2609 = 7*372 + 5; hence not 7 divides 2609 by NAT_4:9;
    2609 = 11*237 + 2; hence not 11 divides 2609 by NAT_4:9;
    2609 = 13*200 + 9; hence not 13 divides 2609 by NAT_4:9;
    2609 = 17*153 + 8; hence not 17 divides 2609 by NAT_4:9;
    2609 = 19*137 + 6; hence not 19 divides 2609 by NAT_4:9;
    2609 = 23*113 + 10; hence not 23 divides 2609 by NAT_4:9;
    2609 = 29*89 + 28; hence not 29 divides 2609 by NAT_4:9;
    2609 = 31*84 + 5; hence not 31 divides 2609 by NAT_4:9;
    2609 = 37*70 + 19; hence not 37 divides 2609 by NAT_4:9;
    2609 = 41*63 + 26; hence not 41 divides 2609 by NAT_4:9;
    2609 = 43*60 + 29; hence not 43 divides 2609 by NAT_4:9;
    2609 = 47*55 + 24; hence not 47 divides 2609 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2609 & n is prime
  holds not n divides 2609 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
