On finiteness of odd superperfect numbers

TL;DR

This paper proves new results about the equation involving the sum-of-divisors function, leading to the conclusion that only finitely many odd superperfect numbers exist with a fixed number of prime factors.

Contribution

It establishes the finiteness of odd superperfect numbers with a given number of prime factors based on new results related to divisor sum equations.

Findings

Finiteness of odd superperfect numbers with fixed prime factors

New results on the equation involving sum-of-divisors functions

Corollary linking divisor equations to superperfect number finiteness

Abstract

Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.