Are All Perfect Numbers Even?/Progress/Form

Theorem
An odd perfect number $n$ is of the form:
 * $n = p^a q^b r^c \cdots$

where:
 * $p, q, r, \ldots$ are prime numbers of the form $4 k + 1$ for some $k \in \Z_{>0}$
 * $a$ is also of the form $4 k + 1$ for some $k \in \Z_{>0}$
 * $b, c, \ldots$ are all even.