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.

Proof

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Historical Note

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

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $28$