BMO Round 1, 2025

Number theory Difficulty 6.0 National Olympiad Prove it United Kingdom

Let $p$ be a prime number, and let $n$ be the smallest positive integer, strictly greater than $1$ , for which $n^6 - 1$ is divisible by $p$ .

Prove that at least one of $(n+1)^6 - 1$ and $(n+2)^6 - 1$ is divisible by $p$ .

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