Some Extensions to Touchard's Theorem on Odd Perfect Numbers

TL;DR

This paper extends Touchard's theorem on the additive structure of odd perfect numbers, providing new congruence conditions and exploring their implications for the form and properties of such numbers.

Contribution

It introduces new congruence-based extensions to Touchard's theorem, broadening understanding of the structure of odd perfect numbers.

Findings

Extended Touchard's result to x=3 and x=α+2 with prime α

Established conditions under which certain congruences hold for odd perfect numbers

Identified that π ≡ 1 mod 12 suffices under specific conditions

Abstract

The multiplicative structure of an odd perfect number $n$, if any, is $n=\pi^\alpha M^2$, where $\pi$ is prime, $\gcd(\pi,M)=1$ and $\pi\equiv \alpha\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv \pi\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $\pi$). We further extend the proof to $x=\alpha+2$, $\alpha+2$ prime, with the restriction that the congruence modulo $4 x$ does not include $n$. Besides, we note that the first extension of Touchard's result holds also with an exclusive disjunction, so that $\pi\equiv 1\pmod{12}$ is a sufficient condition because $3\nmid n$.