
theorem
  8537 is prime
proof
  now
    8537 = 2*4268 + 1; hence not 2 divides 8537 by NAT_4:9;
    8537 = 3*2845 + 2; hence not 3 divides 8537 by NAT_4:9;
    8537 = 5*1707 + 2; hence not 5 divides 8537 by NAT_4:9;
    8537 = 7*1219 + 4; hence not 7 divides 8537 by NAT_4:9;
    8537 = 11*776 + 1; hence not 11 divides 8537 by NAT_4:9;
    8537 = 13*656 + 9; hence not 13 divides 8537 by NAT_4:9;
    8537 = 17*502 + 3; hence not 17 divides 8537 by NAT_4:9;
    8537 = 19*449 + 6; hence not 19 divides 8537 by NAT_4:9;
    8537 = 23*371 + 4; hence not 23 divides 8537 by NAT_4:9;
    8537 = 29*294 + 11; hence not 29 divides 8537 by NAT_4:9;
    8537 = 31*275 + 12; hence not 31 divides 8537 by NAT_4:9;
    8537 = 37*230 + 27; hence not 37 divides 8537 by NAT_4:9;
    8537 = 41*208 + 9; hence not 41 divides 8537 by NAT_4:9;
    8537 = 43*198 + 23; hence not 43 divides 8537 by NAT_4:9;
    8537 = 47*181 + 30; hence not 47 divides 8537 by NAT_4:9;
    8537 = 53*161 + 4; hence not 53 divides 8537 by NAT_4:9;
    8537 = 59*144 + 41; hence not 59 divides 8537 by NAT_4:9;
    8537 = 61*139 + 58; hence not 61 divides 8537 by NAT_4:9;
    8537 = 67*127 + 28; hence not 67 divides 8537 by NAT_4:9;
    8537 = 71*120 + 17; hence not 71 divides 8537 by NAT_4:9;
    8537 = 73*116 + 69; hence not 73 divides 8537 by NAT_4:9;
    8537 = 79*108 + 5; hence not 79 divides 8537 by NAT_4:9;
    8537 = 83*102 + 71; hence not 83 divides 8537 by NAT_4:9;
    8537 = 89*95 + 82; hence not 89 divides 8537 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8537 & n is prime
  holds not n divides 8537 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
