Suppose $m,n$ are positive integers such that $a-b|a^m-b^n , \forall a,b \in \mathbb Z , a-b \ne0$ , then is it true that $m=n$?

Question

Suppose $m,n$ are positive integers such that for all $a\neq b$ one has $a-b\mid a^m-b^n$, then is it true that $m=n$ ?

Answer 1

Hint: You need to show that the above condition implies $2^m-2^n\equiv 0\pmod d$ for all $d$. So $2^m=2^n$.