EDIT, May 2015: in the second edition of the relevant book, the question was corrected, a single number had been mis-typed. The corrected question (thanks to Max) is to find all positive integer pairs such that
$$ m^2 – 1 \; | \; 3^m + \left( n! – 2 \right)^m. $$
$$ $$
The version that caused me all that misery was $ m^2 – 1 \; | \; 3^m + \left( n! – 1 \right)^m. $
EDIT, 2010: it turns out that no answer to this is known, as the authors of the book it is in have now confirmed they do not know how to do it. Will Jagy.
ORIGINAL: I have been wondering if there exist some general techniques to attack problems in which a polynomial divides and exponential equation. The motivation of this question came when trying to solve the following problem:
Find all positive integers "$m$ such that $m^2-1$ divides $3^m+5^m$
I haven't been able to come with a proof, and I am really considering the possibility that it cannot be solved using "elementary" methods.
I would really aprecciate some references (if there are any) to the general question as well as the solution to this very particular case.
Best Answer
The source of the question is "Problems from the Book" by Andreescu and Dospinescu. I finally emailed Andreescu yesterday asking what was going on. He apologised---he says there's a typo in the book. He says he doesn't know how to answer the question. So I think the question should currently be regarded as an open problem. I'll remark that I made a comment under the original question which as I write has 11 upvotes and now has a serious chance of being false ;-)