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): $28$