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For p a prime and n a positive integer, consider the group of units, $(\mathbb{Z}/(p^{n}-1)\mathbb{Z})^{\times}$. How can I go about to find the order of $\bar{p}$?

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    I've made my comment into an answer.2011-09-28

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Clearly, $p^n\equiv 1\bmod (p^n-1)$, so the order of $p$ is $\leq n$. But if $p^k\equiv 1\bmod (p^n-1)$, then $(p^n-1)\mid (p^k-1)$ and if $k then $p^k-1, so this is impossible. Therefore the order must be precisely $n$.

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    @El G: I demonstrate that $p^k\equiv 1\bmod (p^n-1)\implies k\geq n$, using a proof by contradiction (if we had k, we would get the contradiction that $p^n-1\leq p^k-1$ and p^k-1).2011-09-28