How prove this diophantine equation $3^x-2^y=k$ have finitely many integral solutions

Question

For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions.

My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer solution,

But for $k=2m+1$, I can't prove three finitely many integral solution.

Thank you.

Answer 1

This is a result of a theorem due to Carl Størmer. His theorem can be stated as follows:

Let $S=\left\{ p_1^{e_1} p_2^{e_2} \cdots p_j^{e_i} \mid e_i \in \mathbb{N}_0 \right\}$, where $p_1$, $p_2 \cdots p_j$ is a finite collection of prime numbers. Then there are a finite number of pairs $x,y \in S$ such that $x-y=k$ (for fixed $k$).

Unfortunately, the wikipedia page for the theorem only considers the $k=1$ case, when in fact Størmer had proven it for all integer values of $k$.

Sources:

1. http://projecteuclid.org/download/pdf_1/euclid.ijm/1256067456

2. http://en.wikipedia.org/wiki/Størmer's_theorem