
theorem
  8161 is prime
proof
  now
    8161 = 2*4080 + 1; hence not 2 divides 8161 by NAT_4:9;
    8161 = 3*2720 + 1; hence not 3 divides 8161 by NAT_4:9;
    8161 = 5*1632 + 1; hence not 5 divides 8161 by NAT_4:9;
    8161 = 7*1165 + 6; hence not 7 divides 8161 by NAT_4:9;
    8161 = 11*741 + 10; hence not 11 divides 8161 by NAT_4:9;
    8161 = 13*627 + 10; hence not 13 divides 8161 by NAT_4:9;
    8161 = 17*480 + 1; hence not 17 divides 8161 by NAT_4:9;
    8161 = 19*429 + 10; hence not 19 divides 8161 by NAT_4:9;
    8161 = 23*354 + 19; hence not 23 divides 8161 by NAT_4:9;
    8161 = 29*281 + 12; hence not 29 divides 8161 by NAT_4:9;
    8161 = 31*263 + 8; hence not 31 divides 8161 by NAT_4:9;
    8161 = 37*220 + 21; hence not 37 divides 8161 by NAT_4:9;
    8161 = 41*199 + 2; hence not 41 divides 8161 by NAT_4:9;
    8161 = 43*189 + 34; hence not 43 divides 8161 by NAT_4:9;
    8161 = 47*173 + 30; hence not 47 divides 8161 by NAT_4:9;
    8161 = 53*153 + 52; hence not 53 divides 8161 by NAT_4:9;
    8161 = 59*138 + 19; hence not 59 divides 8161 by NAT_4:9;
    8161 = 61*133 + 48; hence not 61 divides 8161 by NAT_4:9;
    8161 = 67*121 + 54; hence not 67 divides 8161 by NAT_4:9;
    8161 = 71*114 + 67; hence not 71 divides 8161 by NAT_4:9;
    8161 = 73*111 + 58; hence not 73 divides 8161 by NAT_4:9;
    8161 = 79*103 + 24; hence not 79 divides 8161 by NAT_4:9;
    8161 = 83*98 + 27; hence not 83 divides 8161 by NAT_4:9;
    8161 = 89*91 + 62; hence not 89 divides 8161 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8161 & n is prime
  holds not n divides 8161 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
