Some Modular Considerations Regarding Odd Perfect Numbers

EasyChair Preprint 2814

Abstract

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $\sigma(m^2)/p^k$ is a square.

Keyphrases: Deficiency, Odd perfect number, Special prime, Sum of aliquot divisors, Sum of divisors

@booklet{EasyChair:2814,
  author    = {Jose Arnaldo Bebita Dris and Immanuel Tobias San Diego},
  title     = {Some Modular Considerations Regarding Odd Perfect Numbers},
  howpublished = {EasyChair Preprint 2814},
  year      = {EasyChair, 2020}}