Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that (a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and (a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.
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Tags: number theory
Let $a$ and $m$ be positive integers and $p$ be an odd prime number such that $p^m\mid a-1$ and $p^{m+1}\nmid a-1$. Prove that (a) $p^{m+n}\mid a^{p^n}-1$ for all $n\in\mathbb N$, and (a) $p^{m+n+1}\nmid a^{p^n}-1$ for all $n\in\mathbb N$.