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- Jan 26, 2012

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Let $n = pq$ such that $p$ and $q$ are distinct primes. Let $a$ be coprime to $n$. Show that the following holds:

$$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$

$$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$

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