Bounds On Factors Of Odd Perfect Numbers

TL;DR

This paper investigates properties of odd perfect numbers, especially the sums of reciprocals of their factors, and establishes bounds on their prime factors based on the number of distinct primes.

Contribution

It provides new bounds on prime factors of odd perfect numbers and analyzes properties of reciprocal sums of their factors, advancing understanding of their structure.

Findings

Established bounds on prime factors of odd perfect numbers.

Analyzed properties of sums of reciprocals of factors.

Provided new insights into the structure of odd perfect numbers.

Abstract

Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the sums of reciprocals of the factors of odd perfect numbers, or, in more technical terms, the properties of the sub-sums of \sigma_{-1} (n). By this result, it also establishes strong bounds on the prime factors of odd perfect numbers using the number of distinct prime factors it may possess.