RO-TST 2004 Problem 13

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$ , we have $a^m - 1\equiv 0 \pmod n$ .

Prove that $n \leq 4m(2^m-1)$ .

Created by Harazi, modified by Marian Andronache.