Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime?
In other words, can a Mersenne number be a Fermat pseudo prime to the base 3?
I tried this but couldn't make much progress. I would appreciate if the solution uses only elementary tools, but any solution is better than none.
Thanks!