I did a little research on odd perfect numbers. We all know that Euclid gave a method for constructing perfect numbers which applies only to even perfect numbers. However, anotehr famous mathematician, Descartes, wrote a letter to Mersenne (I think this is interesting since we learned perfect numbers seem to be in the form of Mersenne Primes) proposing that every even perfect number is of Euclid's form, but that he saw no reason why an odd perfect number could not exist. Descartes was therefore among the first to consider the existence of odd perfect numbers. I read that other mathematicians believed all perfect numbers to be even, even without mathematical proof. They believed all would fall into Euler's theorem, even though Euler also considered odd perfect numbers.

I also found that Euler said if an odd, perfect number exists, it must be in the form,

$N= p^{4x + 1}Q^{2}$ , where p is a prime in the form 4p+1.

My question is, if they don't know if there even exists odd perfect numbers, how do that know what form they would need to be in?