More on the Nonexistence of Odd Perfect Numbers of a Certain Form

Abstract

Euler showed that if an odd perfect number exists, it must be of the form $N=p^{\alpha }{q}_1^{2{\beta }_1}$ $\dots$ $q_k^{2{\beta }_k}$, where $p,q_1,\dots ,q_k$ are distinct odd primes, $\alpha$, ${\beta }_i\ge 1$, for $1\le i\le k$, with $p\equiv \alpha \equiv 1(\mathrm{mod}4)$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each ${\beta }_i\equiv 2(\mathrm{mod}5)$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each ${\beta }_i\equiv 2(\mathrm{mod}5)$.

9 pages