When does $m$ divide $a^m$?

Question

Let $a\ge 0$, $m\ge 1$ be integers. What can be said about $m|a^m$? I note that if $a=1$, then $m\not{|} a^m$ unless $m=1$ and if $a=0$, then always $m|a^m$. Are there any general results for the less trivial cases?

In particular, for any $a\ge 0$ fixed, does there exist $m\ge 1$ such that $m|a^m$?

Answer 1

Hint: Check out Fermat's Little theorem and Euler totient function