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 Th66:
  not 8 divides 3|^n+1
  proof
    per cases;
    suppose n is odd;
      then consider k such that
A1:   n = 2*k+1 by ABIAN:9;
A2:   3|^(2*k+1) = 3|^(2*k)*3 by NEWTON:6;
      3|^(2*k),1 are_congruent_mod 8 by Th65;
      then 3|^(2*k)*3,1*3 are_congruent_mod 8 by Lm14,INT_1:18;
      then 3|^(2*k+1)+1,4 are_congruent_mod 8 by A2;
      then (3|^n+1) mod 8 = 4 mod 8 by A1,NAT_D:64
      .= 4 by NAT_D:24;
      hence thesis by INT_1:62;
    end;
    suppose n is even;
      then consider k such that
A3:   n = 2*k;
      3|^(2*k),1 are_congruent_mod 8 by Th65;
      then 3|^(2*k)+1,1+1 are_congruent_mod 8;
      then (3|^n+1) mod 8 = 2 mod 8 by A3,NAT_D:64
      .= 2 by NAT_D:24;
      hence thesis by INT_1:62;
    end;
  end;
