A new approach to odd perfect numbers via GCDs

TL;DR

This paper introduces a novel GCD-based framework for analyzing odd perfect numbers, deriving key relationships among GCDs of divisor sums and prime powers, and explores their implications for the structure and rarity of such numbers.

Contribution

It establishes a new GCD relation involving odd perfect numbers, provides explicit formulas for these GCDs, and conjectures the scarcity of numbers satisfying certain GCD conditions.

Findings

Proves that G × H = I^2 for specific GCDs in odd perfect numbers.

Derives formulas for G, H, and I in terms of divisor sums and gcds.

Conjectures that the set of numbers with equal GCDs has asymptotic density zero.

Abstract

Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,\sigma(n^2)\bigg)$$ and $$I = \gcd\bigg(n,\sigma(n^2)\bigg).$$ We prove that $G \times H = I^2$. (Note that it is trivial to show that $G \mid I$ and $I \mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $\sigma(q^k)/2, n,$ and $\gcd\bigg(\sigma(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$, then $\sigma(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\mathscr{A} = \{m : \gcd(m,\sigma(m^2))=\gcd(m^2,\sigma(m^2))\}$$ has asymptotic density zero.