Generalized Mersenne Numbers of the form $c{x}^2$

Abstract

Generalized Mersenne numbers are defined as $M_{p,n}=p^n-p+1$, where $p$ is any prime and $n$ is any positive integer. Here, we prove that for each pair $(c,p)$ with $c\ge 1$ an integer, there is at most one $M_{p,n}$ of the form $c{x}^2$ with a few exceptions.

5 pages; to appear in `Annales Mathematicae et Informaticae'