Are All Perfect Numbers Even?

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Open Question

By the Theorem of Even Perfect Numbers, it is known that an even number is perfect if and only if it is of the form:

$2^{n - 1} \paren {2^n - 1}$

where $2^n - 1$ is prime.


It is not known whether there exist any odd perfect numbers.

None have ever been found.


Progress

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

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