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 $\pi\equiv \alpha\equiv1\pmod{4}$. 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 $3\nmid n$.