reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th21:
  31 divides 2|^341-2
  proof
A1: (2|^5)|^68 = 2|^(5*68) by NEWTON:9;
A2: 1 mod 31 = 1 by NAT_D:24;
    2|^5 = 2*2*2*2*2 by Th1
    .= 31*1+1;
    then 2|^5 mod 31 = 1 by NAT_D:def 2;
    then (2|^5)|^68 mod 31 = 1 by NEWTON05:15;
    then
A3: 2|^340,1 are_congruent_mod 31 by A1,A2,NAT_D:64;
    2|^(340+1)-2 = (2|^340*2|^1)-(2*1) by NEWTON:8
    .= 2*(2|^340-1);
    hence thesis by A3,INT_2:2;
  end;
