Upper bounds on the second largest prime factor of an odd perfect number

TL;DR

This paper improves upper bounds on the second largest prime factor of an odd perfect number, refining previous results and providing tighter constraints based on the relationships between the prime factors.

Contribution

It introduces new bounds on the second largest prime factor of odd perfect numbers, extending prior work with novel inequalities and conditions for prime factor proximity.

Findings

$p_{k-1} < (2N)^{1/5}$

$p_{k-1}p_k < 6^{1/4}N^{1/2}$

Bounds can be strengthened if $p_k$ and $p_{k-1}$ are close

Abstract

Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that $p_{k-1}< (2N)^{1/5}$ and that $p_{k-1}p_k < 6^{1/4}N^{1/2}.$ We also show that if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further strengthened.