Some Extensions to Touchard's Theorem on Odd Perfect Numbers

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 $\pi \equiv \alpha \equiv 1(\mathrm{mod}4)$. An additive structure of $n$, established by Touchard, is that "$(n\equiv 9(\mathrm{mod}36))$ OR $(n\equiv 1(\mathrm{mod}12))$". A first extension of Touchard's result is that the proposition "$(n\equiv x^2(\mathrm{mod}4x^2))$ OR $(n\equiv \pi \equiv 1(\mathrm{mod}4x))$" 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 $4x$ does not include $n$. Besides, we note that the first extension of Touchard's result holds also with an exclusive disjunction, so that $\pi \equiv 1(\mathrm{mod}12)$ is a sufficient condition because $3\nmid n$.

5 pages