reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th72:
  n is even & 3|^n - 2|^m = 1 implies n = 2 & m = 3
  proof
    assume that
A1: n is even and
A2: 3|^n - 2|^m = 1;
    consider k such that
A3: n = 2*k by A1;
A4: 3|^(2*k) = 3|^k|^2 by NEWTON:9
    .= (3|^k)^2 by WSIERP_1:1;
A5: 2|^m = 3|^n - 1 by A2;
    3|^k+1 divides (3|^k+1)*(3|^k-1);
    then consider a such that
A6: 3|^k+1 = 2|^a by A5,A3,A4,NEWTON03:36,XPRIMES1:2;
    3|^k-1 divides (3|^k+1)*(3|^k-1);
    then consider b such that
A7: 3|^k-1 = 2|^b by A5,A3,A4,NEWTON03:36,XPRIMES1:2;
    2|^a - 2|^b = 2 by A6,A7;
    then 3|^k+1 = 4 by A6,Lm3,Th70;
    then 3|^k = 3|^1;
    then k = 1 by PEPIN:30;
    hence thesis by A3,A5,Lm8,Lm12,PEPIN:30;
  end;
