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Any ideas on how to solve the congruences \begin{eqnarray*} p^k &\equiv& 1 \mod q \\ q &\equiv& 1 \mod p \end{eqnarray*} where $p$ and $q$ are primes and $k$ is a positive integer?

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    $(p,q,k)=(2,7,3)$ is one solution. What do you want - all solutions? There might be quite a few....2012-08-15
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    Indeed, every Mersenne prime $q=2^k-1$ will give a solution with $p=2$, and it's conjectured there are infinitely many. That gives you at least 47 solutions, anyway.2012-08-15
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    Alternatively, for every pair of primes $p,q$ such that $p$ divides $q-1$, we can take $k$ to be $q-1.$ This gives a ton of solutions...2012-08-15

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