Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Prove it Saudi Arabia

Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides
$1^{n}+2^{n}+\cdots+(p+1)^{n}$

Solution

Let $k$ be a positive integer. Using Fermat's little theorem we have
$1^{k(p-1)}+2^{k(p-1)}+\cdots+(p+1)^{k(p-1)} \equiv \underbrace{1^{k}+1^{k}+\cdots+1^{k}}_{p-1 \text{ times}}+0^{k}+1^{k} \equiv 0 \pmod{p} .$
Therefore, for $n=k(p-1)$ , and $k=1,2,3, \ldots$ , the prime number $p$ divides
$1^{n}+2^{n}+\cdots+(p+1)^{n} .$

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