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
its greatest prime factor is greater than $300 \, 000$
its second largest prime factor is greater than $1000$
at least one of the prime powers factoring it is greater than $10^{18}$
if less than $10^{9118}$ then it is divisible by the $6$th power of some prime.


Proof


Sources