Problem

Source: Turkey JBMO TST 2016 P3

Tags: number theory, prime numbers



Let $n$ be a positive integer, $p$ and $q$ be prime numbers such that \[ pq \mid n^p+2 \quad \text{and} \quad n+2 \mid n^p+q^p. \]Prove that there exists a positive integer $m$ satisfying $q \mid 4^m \cdot n +2$.