Generalized Mersenne Numbers of the form $cx^2$

TL;DR

This paper investigates the properties of generalized Mersenne numbers, proving that for most cases, there is at most one such number of the form $cx^2$, with some exceptions, contributing to number theory and prime-related sequences.

Contribution

It establishes a uniqueness result for generalized Mersenne numbers of the form $cx^2$, extending understanding of their structure and distribution.

Findings

Most generalized Mersenne numbers of the form $cx^2$ are unique for given $(c,p)$

There are only a few exceptions to the uniqueness result

The paper advances knowledge on the intersection of prime powers and quadratic forms

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 $c\geq 1$ an integer, there is at most one $M_{p, n}$ of the form $cx^2$ with a few exceptions.