An exponential diophantine equation related to odd perfect numbers

TL;DR

This paper investigates a specific exponential Diophantine equation linked to odd perfect numbers, establishing upper bounds on solutions and exploring implications for the longstanding odd perfect number problem.

Contribution

It proves that the equation has at most four positive solutions for certain primes and applies this result to the study of odd perfect numbers.

Findings

At most four solutions for the Diophantine equation under given conditions

New bounds and constraints related to odd perfect numbers

Application of the equation's properties to number theory problems

Abstract

We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.