Divisibility of generalized Mersenne numbers

TL;DR

This paper establishes a divisibility criterion for generalized Mersenne numbers, which are repunits in different bases, providing both a general and an alternative proof for a special case.

Contribution

It introduces a new theorem that determines when two generalized Mersenne numbers are divisible, expanding understanding of their properties in different bases.

Findings

A criterion for divisibility of generalized Mersenne numbers is proven.

An alternative proof is provided for a special case.

The results enhance the theoretical understanding of repunits in various bases.

Abstract

In this article, I present a theorem determining a criterion for divisibility of two generalized Mersenne numbers, which are repunits of the same length in base-$a^m$ and base-$a^k$. In addition to the general proof, I present an alternative proof for a special case of the theorem.