# Are All Perfect Numbers Even?/Progress/Minimum Size

## Theorem

It had been established by $1986$ that an odd perfect number, if one were to exist, would have over $200$ digits.

By $1997$ that lower bound had been raised to $300$ digits.

By $2012$ that lower bound had been raised again to $1500$ digits.

## Proof

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## Historical Note

Bryant Tuckerman published a proof in $1968$ that an odd perfect number is greater than $10^{36}$.

Peter Hagis, Jr. published a proof in $1973$ that an odd perfect number is greater than $10^{50}$.

Richard P. Brent and Graeme L. Cohen published a proof in $1989$ that an odd perfect number is greater than $10^{160}$.

Richard P. Brent, Graeme L. Cohen and Hermanus Johannes Joseph te Riele published a proof in $1991$ that an odd perfect number is greater than $10^{300}$.

Pascal Ochem and Michaël Rao published a proof in $2012$ that an odd perfect number is greater than $10^{1500}$.

## Sources

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