Let $1 + 2^m = 3^n$. What the max value of $(m+n)$?

Question

How do I determine the maximum value of $(m+n)$ if $m$ and $n$ are natural numbers if $1 + 2^m = 3^n$ holds? I have got $\text {max} (m+n)$ to be $5$ so far, but I do not know how to determine whether there is a larger possible sum. Help is much appreciated!

Answer 1

To show that there are no bigger solutions, suppose that $m\ge 2$. Then $3^n\equiv 1\pmod{4}$, and therefore $n$ must be even, say $n=2k$. We then have $$2^m=(3^k-1)(3^k+1).$$ Thus $3^k-1$ and $3^k+1$ are powers of $2$. The only two powers of $2$ that differ by $2$ are $2$ and $4$. It follows that $3^k-1=2$, and therefore $k=1$.