Find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$ 
As is the question in the title, I am wishing to find all positive integers $n$ such that $3^n + 5^n$ is divisible by $n^2 - 1$. 

I have so far shown both expressions are divisible by $8$ for odd $n\ge 3$ so trivially a solution is $n=3$. I'm not quite sure how to proceed now though. I have conjectured that the only solution is $n=3$ and have tried to prove it but have had little luck. Can anyone point me in the right direction? thanks

I'd better make this an answer. This was asked on MO long ago. Nobody could do it. Kevin Buzzard wrote to Andreescu and found out that the authors of the book don't know how to finish the problem. I put an answer summarizing what we had, in basic language. 
See
https://mathoverflow.net/questions/16341/on-polynomials-dividing-exponentials
