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

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## Theorem

An odd perfect number has:

- at least $8$ distinct prime factors
- at least $11$ distinct prime factors if $3$ is not one of them
- at least $101$ prime factors (not necessarily distinct)
- its greatest prime factor is greater than $1 \, 000 \, 000$
- its second largest prime factor is greater than $1000$
- at least one of the prime powers factoring it is greater than $10^{62}$
- if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.

## Proof

## Historical Note

James Joseph Sylvester stated that there exist no odd perfect number with fewer than $6$ distinct prime factors, and proved that there are none with fewer than $8$ if none of those prime factors is $3$.

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

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