# Are All Perfect Numbers Even?/Progress/Prime Factors/Historical Note

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## Historical Note on Prime Factors of Odd Perfect Number

Bryant Tuckerman published a proof in $1973$ that an odd perfect number $P$ has the properties that:

- either:
- at least one of the prime powers factoring $P$ is greater than $10^{18}$
- the power of such a prime factor is even

- or:
- there is no divisor of $P$ less than $7$.

Peter Hagis, Jr. published a proof in $1980$ that an odd perfect number has at least $8$ distinct prime factors.

Peter Hagis, Jr. and Graeme L. Cohen published a proof in $1998$ that an odd perfect number has at least one prime factor which is greater than $1 \, 000 \, 000$.

Pascal Ochem and Michaël Rao published a proof in $1998$ that:

- at least one of the prime powers factoring an odd perfect number is greater than $10^{62}$
- an odd perfect number has more than $101$ prime factor (not neessarily distinct).

## Sources

- 1973: Bryant Tuckerman:
*A Search Procedure and Lower Bound for Odd Perfect Numbers*(*Math. Comp.***Vol. 27**: 943 – 949) www.jstor.org/stable/2005529

- 1980: Peter Hagis, Jr.:
*An Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors*(*Math. Comp.***Vol. 34**: 1027 – 1032) www.jstor.org/stable/2006211

- 1998: Peter Hagis, Jr. and Graeme L. Cohen:
*Every Odd Perfect Number Has a Prime Factor Which Exceeds $10^6$*(*Math. Comp.***Vol. 67**: 1323 – 1330) www.jstor.org/stable/2585187

- 2012: Pascal Ochem and Michaël Rao:
*Odd Perfect Numbers Are Greater than $10^{1500}$*(*Math. Comp.***Vol. 81**: 1869 – 1877) www.jstor.org/stable/23268069