  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
$\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We
then give an application of this result to the case when $\sigma(m^2)/p^k$ is a
square.
