A Partial Proof of a Conjecture of Dris

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, $q\equiv k\equiv 1(\mathrm{mod}4)$, 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.

7 pages