Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.
Source: MOP 2006 Homework - Black Group
Tags: number theory unsolved, number theory
Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.
