
theorem
  6827 is prime
proof
  now
    6827 = 2*3413 + 1; hence not 2 divides 6827 by NAT_4:9;
    6827 = 3*2275 + 2; hence not 3 divides 6827 by NAT_4:9;
    6827 = 5*1365 + 2; hence not 5 divides 6827 by NAT_4:9;
    6827 = 7*975 + 2; hence not 7 divides 6827 by NAT_4:9;
    6827 = 11*620 + 7; hence not 11 divides 6827 by NAT_4:9;
    6827 = 13*525 + 2; hence not 13 divides 6827 by NAT_4:9;
    6827 = 17*401 + 10; hence not 17 divides 6827 by NAT_4:9;
    6827 = 19*359 + 6; hence not 19 divides 6827 by NAT_4:9;
    6827 = 23*296 + 19; hence not 23 divides 6827 by NAT_4:9;
    6827 = 29*235 + 12; hence not 29 divides 6827 by NAT_4:9;
    6827 = 31*220 + 7; hence not 31 divides 6827 by NAT_4:9;
    6827 = 37*184 + 19; hence not 37 divides 6827 by NAT_4:9;
    6827 = 41*166 + 21; hence not 41 divides 6827 by NAT_4:9;
    6827 = 43*158 + 33; hence not 43 divides 6827 by NAT_4:9;
    6827 = 47*145 + 12; hence not 47 divides 6827 by NAT_4:9;
    6827 = 53*128 + 43; hence not 53 divides 6827 by NAT_4:9;
    6827 = 59*115 + 42; hence not 59 divides 6827 by NAT_4:9;
    6827 = 61*111 + 56; hence not 61 divides 6827 by NAT_4:9;
    6827 = 67*101 + 60; hence not 67 divides 6827 by NAT_4:9;
    6827 = 71*96 + 11; hence not 71 divides 6827 by NAT_4:9;
    6827 = 73*93 + 38; hence not 73 divides 6827 by NAT_4:9;
    6827 = 79*86 + 33; hence not 79 divides 6827 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 6827 & n is prime
  holds not n divides 6827 by XPRIMET1:44;
  hence thesis by NAT_4:14;
end;
