# Are All Perfect Numbers Even?/Progress/Form

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## Contents

## 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

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