
theorem
  8609 is prime
proof
  now
    8609 = 2*4304 + 1; hence not 2 divides 8609 by NAT_4:9;
    8609 = 3*2869 + 2; hence not 3 divides 8609 by NAT_4:9;
    8609 = 5*1721 + 4; hence not 5 divides 8609 by NAT_4:9;
    8609 = 7*1229 + 6; hence not 7 divides 8609 by NAT_4:9;
    8609 = 11*782 + 7; hence not 11 divides 8609 by NAT_4:9;
    8609 = 13*662 + 3; hence not 13 divides 8609 by NAT_4:9;
    8609 = 17*506 + 7; hence not 17 divides 8609 by NAT_4:9;
    8609 = 19*453 + 2; hence not 19 divides 8609 by NAT_4:9;
    8609 = 23*374 + 7; hence not 23 divides 8609 by NAT_4:9;
    8609 = 29*296 + 25; hence not 29 divides 8609 by NAT_4:9;
    8609 = 31*277 + 22; hence not 31 divides 8609 by NAT_4:9;
    8609 = 37*232 + 25; hence not 37 divides 8609 by NAT_4:9;
    8609 = 41*209 + 40; hence not 41 divides 8609 by NAT_4:9;
    8609 = 43*200 + 9; hence not 43 divides 8609 by NAT_4:9;
    8609 = 47*183 + 8; hence not 47 divides 8609 by NAT_4:9;
    8609 = 53*162 + 23; hence not 53 divides 8609 by NAT_4:9;
    8609 = 59*145 + 54; hence not 59 divides 8609 by NAT_4:9;
    8609 = 61*141 + 8; hence not 61 divides 8609 by NAT_4:9;
    8609 = 67*128 + 33; hence not 67 divides 8609 by NAT_4:9;
    8609 = 71*121 + 18; hence not 71 divides 8609 by NAT_4:9;
    8609 = 73*117 + 68; hence not 73 divides 8609 by NAT_4:9;
    8609 = 79*108 + 77; hence not 79 divides 8609 by NAT_4:9;
    8609 = 83*103 + 60; hence not 83 divides 8609 by NAT_4:9;
    8609 = 89*96 + 65; hence not 89 divides 8609 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8609 & n is prime
  holds not n divides 8609 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
