Could someone please explain why it is still on hold. The question is pretty clear now, at least I think so

Thanks you for the answer but the problem still remains of finding all $s$-gonal numbers that are Mersenne for a specific $s$. For example if $s$=82 (random value I made up), how can I find all of the solutions $n,p$ or prove that there are none. But not by tackling every $s$ by itself and using a different method for each one. The Mordell equation way that I mentioned was one way of doing it, but since calculating for large $k$ is very hard, I want to come up with another (easier) way of doing this