Is the conjecture true? $3^n - 2^m = 1$ has infinitely many solutions, where n, m are natural numbers.

Question

Is the conjecture true?

$3^n - 2^m = 1 $ has infinitely many solutions, where n, m are natural numbers.

More generally, $P^n-Q^m=1$ has infinitely many solution for n, m ϵ {1,2,3,…} where P is odd, Q is even, P and Q have no common factors.

$P^n-Q^m=2$ has infinitely many solution for n, m ϵ {1,2,3,…} where P, Q are both odd, P and Q have no common factors.

This is my first post - I joined this community to look into this.

Answer 1

Mihăilescu's theorem, before 2002 known as Catalan's conjecture, states that:

The only natural numbers $m,n>1$ and $P,Q>0$ satisfying $$P^n-Q^m=1,$$are $(m,n,P,Q)=(2,3,3,2)$.