# Are All Perfect Numbers Even?/Progress

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## Progress on Open Question: Are All Perfect Numbers Even?

While it is not known whether there exist any odd perfect numbers, several important facts have been established.

### Minimum Size of Odd Perfect Number

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.

### Form of Odd Perfect Number

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.

### Prime Factors of Odd Perfect Number

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.

## Sources

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