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Hi there, recently I came across the following divisibility question, and I wondered if much can be said about it. Let $p$ and $q$ be different primes, and suppose $p^n + q^r$ divides $p^{2m} - 1$, where $n$, $m$, $r$ are positive integers, $n$ divides $m$, and $q^r > p^m$. Is a classification of the triples $(n,m,r)$ within reach ? Thanks !

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Can you prove that there are infinitely many such triples? The conditions you require are loose enough that it certainly seems like there should be, but I can't think of a proof or come up with an algorithm to generate them. – Dimitrije Kostic Nov 25 '11 at 19:51
There are readily available tabulations of factorizations of numbers of the form $a^t-1$. Might be a good idea to go through those looking for divisors of the desired form to get some idea of what your triples look like. – Gerry Myerson Nov 25 '11 at 21:32

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