TITLE: $3^n - 2^m = \pm 41$ is not possible. How to prove it?
QUESTION [10 upvotes]: $3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?

REPLY [23 votes]: The congruence $3^n - 2^m \equiv 41\pmod{60}$ has no solutions.
The congruence $3^n - 2^m \equiv -41\pmod{72}$ has no solutions.