TITLE: Prove among two integers $2^n-1$ and $2^n+1$, at most one is prime.
QUESTION [6 upvotes]: For any $n>2$, among $2^n-1$ and $2^n+1$, at most one is prime.

I have no clue :(

REPLY [33 votes]: One of the three consecutive integers $(2^n-1)$, $2^n$, and $(2^n+1)$ is a multiple of 3.  It's not $2^n$, so one of $2^n-1\,$ and $2^n+1\, $ is a multiple of three (and $\neq 3$), so at most one of them is prime.