TITLE: Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers $a$ and $b$.
QUESTION [6 upvotes]: Let $p$ be a prime. Prove that $p$ divides $ab^p−ba^p$ for all integers
$a$ and $b$.

REPLY [7 votes]: Hint $\ $ little Fermat $\rm\Rightarrow mod\ p\!:\ n^p \equiv n\ \Rightarrow\ f(a,b^p)\equiv f(a,b)\equiv f(a^p,b)\ $ for all $\rm\:f\in \mathbb Z[x,y]$