Are All Perfect Numbers Even?

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.