On the third largest prime divisor of an odd perfect number

TL;DR

This paper investigates bounds on the third largest prime divisor of odd perfect numbers, providing new inequalities and exploring properties of related prime pairs to advance understanding of their prime factorization structure.

Contribution

It introduces new bounds on prime divisors of odd perfect numbers and analyzes properties of specific prime pairs related to divisor sums, advancing theoretical understanding.

Findings

Proved that the third largest prime divisor a < 2N^{1/6}.

Established that the product of the three largest prime divisors, abc, < (2N)^{3/5}.

Analyzed properties of σ_{m,n} prime pairs to understand divisor relationships.

Abstract

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closely related problem of improving bounds $bc$, and $abc$. In particular, we prove two results. First we prove a new general bound on any prime divisor of an odd perfect number and obtain as a corollary of that bound that $$a < 2N^{\frac{1}{6}}.$$ Second, we show that $$abc < (2N)^{\frac{3}{5}}.$$ We also show how in certain circumstances these bounds and related inequalities can be tightened. Define a $\sigma_{m,n}$ pair to be a pair primes $p$ and $q$ where $q|\sigma(p^m)$, and $p|\sigma(q^n)$. Many of our results revolve around understanding $\sigma_{2,2}$ pairs. We also prove results concerning $\sigma_{m,n}$ pairs for other values of $m$ and $n$.