If we suppose, that a perfect odd number exists, can we determine whether infinite many exist?

It is currently not known whether odd perfect numbers (numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$, including $1$ and $n$) exist.

But suppose, a perfect odd number exists.

Do we then know whether there are infinite many perfect odd numbers ?

This could be possible because various necessary conditions are known for an odd number to be perfect. Perhaps they allow to construct arbitary many perfect odd numbers, supposing that an odd perfect number exists.

• Are you sure this problem not open ? – SSepehr Jan 27 '17 at 10:59
• @SSepehr No, therefore I ask – Peter Jan 27 '17 at 10:59
• Well, we don't even know whether there are infinite perfect numbers (either even or odd), so I'd say this is very close to be an open question... – DonAntonio Jan 27 '17 at 11:31
• @JoseArnaldoBebitaDris What did you complete ? The proof that there are no perfect numbers, or whether there are infinite many if there is one ? And if it is the latter, are there infinite many or not (assuming that there is one) ? – Peter Sep 8 '17 at 10:50
• @JoseArnaldoBebitaDris Thank you! The EVEN perfect numbers are closely related to the Mersenne prime numbers and it is conjectured (but not proven) that there are infinite many Mersenne prime numbers which would imply that there are infinite many EVEN perfect numbers. – Peter Sep 8 '17 at 11:35