On the Components of an Odd Perfect Number

TL;DR

This paper investigates the structure of odd perfect numbers, establishing bounds on their prime components and ratios, supported by numerical analysis of their abundancy indices, and demonstrates that the square part exceeds a certain enormous threshold.

Contribution

It proves a new inequality relating prime factors of odd perfect numbers and provides numerical bounds on their components and ratios.

Findings

${p^k} < (2/3){m^2}$ for odd perfect numbers with special prime $p$

Numerical bounds on abundancy indices and ratios involving $p^k$ and $m^2$

Demonstrates that $m^2 > rac{ ext{sqrt}(6)}{2} imes 10^{150}$

Abstract

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios $\frac{\sigma(p^k)}{m^2}$ and $\frac{\sigma(m^2)}{p^k}$, are used. It is also showed that $m^2 > \frac{\sqrt{6}}{2}({10}^{150})$.