
theorem
  3637 is prime
proof
  now
    3637 = 2*1818 + 1; hence not 2 divides 3637 by NAT_4:9;
    3637 = 3*1212 + 1; hence not 3 divides 3637 by NAT_4:9;
    3637 = 5*727 + 2; hence not 5 divides 3637 by NAT_4:9;
    3637 = 7*519 + 4; hence not 7 divides 3637 by NAT_4:9;
    3637 = 11*330 + 7; hence not 11 divides 3637 by NAT_4:9;
    3637 = 13*279 + 10; hence not 13 divides 3637 by NAT_4:9;
    3637 = 17*213 + 16; hence not 17 divides 3637 by NAT_4:9;
    3637 = 19*191 + 8; hence not 19 divides 3637 by NAT_4:9;
    3637 = 23*158 + 3; hence not 23 divides 3637 by NAT_4:9;
    3637 = 29*125 + 12; hence not 29 divides 3637 by NAT_4:9;
    3637 = 31*117 + 10; hence not 31 divides 3637 by NAT_4:9;
    3637 = 37*98 + 11; hence not 37 divides 3637 by NAT_4:9;
    3637 = 41*88 + 29; hence not 41 divides 3637 by NAT_4:9;
    3637 = 43*84 + 25; hence not 43 divides 3637 by NAT_4:9;
    3637 = 47*77 + 18; hence not 47 divides 3637 by NAT_4:9;
    3637 = 53*68 + 33; hence not 53 divides 3637 by NAT_4:9;
    3637 = 59*61 + 38; hence not 59 divides 3637 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 3637 & n is prime
  holds not n divides 3637 by XPRIMET1:34;
  hence thesis by NAT_4:14;
end;
