# Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes: reference request or proposal as an exercise

I don't know if the following question is in the literature, please add a commment if it is in the literature (I add my thoughts and motivation below in last paragraph, it is discursive and speculative).

An odd perfect number is an odd integer $$N\geq 1$$ such $$\sigma(N)=\sum_{1\leq d\mid N}d=2N.$$ I add the Wikipedia article for Perfect number.

Question. Is it possible to rule out/discard that the only prime factors of an odd perfect number are (a suitable choice of) Mersenne primes and/or Fermat primes? I'm asking if we can to disprove the existence of odd perfect numbers having as prime divisors exclusively Mersenne primes and Fermat primes (it is unknown if there exist infinitely many Mersenne primes and it is unkonwn if there exists finitely many Fermat primes). Many thanks.

I'm asking it as a reference request to know if this question is in the literature, then refer it or add a comment with the bilbiography and I try to search and read it from the literature. In other case I'm asking about what work can be done for my Question, and after some feedback in answers I should to choose an answer.

My only idea to do some work about the veracity of the question is try to compare to Euler's thorem for odd perfect numbers, and the theory of odd perfect numbers.

I don't know Florian Luca, The anti-social Fermat number, American Mathematical Monthly, 107 (2): pp. 171–173 (2000), I know about it from an informative point of view: what refers the Wikipedia section Other interesting facts from the Wikipedia article Fermat number.

It seems reasonable to think that the problem of the existence of odd perfect numbers is unrelated to the problem concerning even perfect numbers. Then we can to think in my question as a question of miscellany in mathematics. I've persuaded myself about certain things about some unsolved problems in mathematics. I know that this isn't scientific. Thus to avoid these ideas we propose this exercise just as a miscellaneous problem. What I evoke is that a different option that odd perfect numbers are unrelated to certain constellations of primes, is the speculative option that there is a close relationship.

Now this post is crossposted on MathOverflow as MO 362043.

I add the links for this Mathematics Stack Exchange about the posts in which I was inspired.

## References:

[1] Could a Mersenne prime divide an odd perfect number?, MSE 2798459 from this Mathematics Stack Exchange (May 27 '18).

[2] Could a Fermat prime divide an odd perfect number?, MSE 2960850 from this Mathematics Stack Exchange (Oct 18 '18).

• In this site Mathematics Stack Exchange there were the question with identificator 2798459 and title Could a Mersenne prime divide an odd perfect number? (May 27 '18) and the question with identificator 2960850 and title Could a Fermat prime divide an odd perfect number? (Oct 18 '18). Commented Apr 21, 2020 at 12:04
• Also feel free to add your feedback for this post or other of my posts in Mathematics Stack Exchange, many thanks. Commented Apr 21, 2020 at 12:26
• To the moderator team, I've accepted an answer for the linked post on MathOverflow, question with identificator 362043 from MathOverflow. Commented Jun 3, 2020 at 7:24