On the Quantity $m^2 - p^k$ Where $p^k m^2$ is an Odd Perfect Number

EasyChair Preprint 3967

Abstract

We prove that $m^2 - p^k$ is not a square, if $n = p^k m^2$ is an odd perfect number with special prime $p$, under the hypothesis that $\sigma(m^2)/p^k$ is a square. We are also able to prove the same assertion without this hypothesis. We also show that this hypothesis is incompatible with the set of assumptions $$\bigg(m^2 - p^k \text{ is a power of two }\bigg) \land \bigg(p \text{ is a Fermat prime}\bigg).$$ We end by stating some conjectures.

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

@booklet{EasyChair:3967,
  author    = {Jose Arnaldo Dris and Immanuel San Diego},
  title     = {On the Quantity $m^2 - p^k$ Where $p^k m^2$ is an Odd Perfect Number},
  howpublished = {EasyChair Preprint 3967},
  year      = {EasyChair, 2020}}