
theorem :: Problem 136
  for k,m,n being Nat st
   k >= 1 & m = 2 |^ k - 2 & n = (2 |^ k) * (2 |^ k - 2)
    holds
      PrimeDivisors m = PrimeDivisors n &
      PrimeDivisors (m+1) = PrimeDivisors (n+1)
  proof
    let k,m,n be Nat;
    assume
A1: k >= 1 & m = 2 |^ k - 2 & n = (2 |^ k) * (2 |^ k - 2);
A2: PrimeDivisors (n + 1) = PrimeDivisors (2 |^ k - 1) by P136c,A1
         .= PrimeDivisors (m + 1) by A1;
    PrimeDivisors m = {2} \/ PrimeDivisors (2 |^ (k-'1) - 1) by A1,P136b
               .= PrimeDivisors n by P136a,A1;
    hence thesis by A2;
  end;
