A Partial Proof of a Conjecture of Dris

TL;DR

This paper investigates properties of hypothetical odd perfect numbers, proving that the prime component q is less than n and establishing the inequality q^k < n in many cases, advancing understanding of their structure.

Contribution

The paper proves that q < n for all odd perfect numbers and verifies the inequality q^k < n in numerous cases, providing new insights into their form.

Findings

q < n for all odd perfect numbers

q^k < n holds in many cases

Advances understanding of odd perfect number structure

Abstract

Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect numbers. Afterwards, we show $q^k < n$ holds in many cases.