Some Modular Considerations Regarding Odd Perfect Numbers

Let $p^k{m}^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma (m^2)\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ holds if and only if $p\equiv k(\mathrm{m}\mathrm{o}\mathrm{d}8)$. We then give an application of this result to the case when $\sigma (m^2)\mathrm{/}{p}^k$ is a square.