
theorem
  8461 is prime
proof
  now
    8461 = 2*4230 + 1; hence not 2 divides 8461 by NAT_4:9;
    8461 = 3*2820 + 1; hence not 3 divides 8461 by NAT_4:9;
    8461 = 5*1692 + 1; hence not 5 divides 8461 by NAT_4:9;
    8461 = 7*1208 + 5; hence not 7 divides 8461 by NAT_4:9;
    8461 = 11*769 + 2; hence not 11 divides 8461 by NAT_4:9;
    8461 = 13*650 + 11; hence not 13 divides 8461 by NAT_4:9;
    8461 = 17*497 + 12; hence not 17 divides 8461 by NAT_4:9;
    8461 = 19*445 + 6; hence not 19 divides 8461 by NAT_4:9;
    8461 = 23*367 + 20; hence not 23 divides 8461 by NAT_4:9;
    8461 = 29*291 + 22; hence not 29 divides 8461 by NAT_4:9;
    8461 = 31*272 + 29; hence not 31 divides 8461 by NAT_4:9;
    8461 = 37*228 + 25; hence not 37 divides 8461 by NAT_4:9;
    8461 = 41*206 + 15; hence not 41 divides 8461 by NAT_4:9;
    8461 = 43*196 + 33; hence not 43 divides 8461 by NAT_4:9;
    8461 = 47*180 + 1; hence not 47 divides 8461 by NAT_4:9;
    8461 = 53*159 + 34; hence not 53 divides 8461 by NAT_4:9;
    8461 = 59*143 + 24; hence not 59 divides 8461 by NAT_4:9;
    8461 = 61*138 + 43; hence not 61 divides 8461 by NAT_4:9;
    8461 = 67*126 + 19; hence not 67 divides 8461 by NAT_4:9;
    8461 = 71*119 + 12; hence not 71 divides 8461 by NAT_4:9;
    8461 = 73*115 + 66; hence not 73 divides 8461 by NAT_4:9;
    8461 = 79*107 + 8; hence not 79 divides 8461 by NAT_4:9;
    8461 = 83*101 + 78; hence not 83 divides 8461 by NAT_4:9;
    8461 = 89*95 + 6; hence not 89 divides 8461 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8461 & n is prime
  holds not n divides 8461 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
