Are All Perfect Numbers Even?/Progress/Form

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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.


Proof


Historical Note

The form that an odd perfect number would need to take was proved by Leonhard Paul Euler.


Sources