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): $28$