On the quantity $m^2-p^k$ where $p^k m^2$ is an odd perfect number -- Part II

TL;DR

This paper investigates inequalities involving odd perfect numbers of the form $p^k m^2$, establishing conditions under which $m < p^k$ and providing improved bounds for the difference $m^2 - p^k$, advancing understanding of their structure.

Contribution

The authors extend previous results by proving new inequalities and bounds for odd perfect numbers with a special prime, under specific hypotheses.

Findings

Proved that $m < p^k$ under certain conditions involving $m^2 - p^k$

Established that $m^2 - p^k > 2m$ and improved it to $\frac{313}{315}m^2$ unconditionally

Provided new inequalities constraining the structure of odd perfect numbers

Abstract

Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {313m^2}/315$.