"I am sure there are infinitely many perfect numbers"

Question

The question Are there infinitely many perfect numbers? is a classic old unsolved problem. However, we keep finding perfect numbers (via Mersenne primes) and produce a lot of knowledge on perfect numbers, e.g. distribution and more. And when I am reading about this topic perfect numbers are often treated (maybe it is just my perception) as if they were infinitely many. And to be honest: After learning about perfect numbers etc. my intuition now tells me that there have to be infinitely many of them, no?.

I just realized this personal bias and want ask those who are experts in this field: Isn't it more likely that there are infinitely many perfect number? Are there some mathematical statements that make the existence of infinitely many perfect numbers more likely? And: Does it even make sense to ask this question?

$~$

Remark 1: In the book The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike the authors point out various aspects from different mathematical fields that all point to the validity of the Riemann hypothesis. They concluded that it is quite unlikely (whatever this means) for the RH to be false (they emphasized: not because of a lot of empirical data, but because of some other mathematical connections that make the validity more likely). This motivated my questions: So are there some mathematical facts that make the existence of infinitely many perfect numbers more likely?

Remark 2: Sorry for such a provocative title :)

Answer 1

This is not a complete answer. Just my views:

Assuming that you know that a perfect number is the one which is of the form $2^{p-1}\times (2^p - 1)$ where p and $2^p-1$ are primes (the latter being well known Mersenne primes).

Euler established this well known one to one correspond hence between Mersenne primes and perfect numbers. So if you have a Mersenne primes you get an even perfect number and vice versa (it is still unknown whether there are any odd perfect numbers or not).

Sadly, this is the only way of finding perfect numbers. So in order to discuss the infinitude of perfect numbers, you need to ultimately discuss infinitude of Mersenne primes which is not a piece of cake. Many great mathematical minds think that answer for the question are there infinitely many Mersenne primes? is yes!! but again there is no proof yet.

See this

Answer 2

(Note: What follows was previously posted as an answer to this related MSE question, but has now been deleted.)

This is not a direct answer to your question, but it is certainly related.

We do know that $$I(q^k) < \frac{5}{4} < \frac{3}{2} \leq I(2^{p-1})$$ where $N=q^k n^2$ is an odd perfect number in Eulerian form and $M=(2^p - 1){2^{p-1}}$ is an even perfect number in Euclidean form.

What follows are incorrect, as rightly pointed out by reuns. I leave the rest of the original answer untouched, mainly for my own benefit.

Because of a property satisfied by the abundancy index $I(x)=\sigma(x)/x$ at prime powers, it follows that $$2^{p-1} < q^k.$$

It follows that, since $q^k < n^2$ [Dris, 2012], if $N$ is bounded, then $M$ is also bounded.

Thus, if there exists an odd perfect number, we would infer that there are only finitely many even perfect numbers.

This is in response to user243301's other answer to the OP.

These images are from WolframAlpha, and were a result of two (similar yet different) inequalities. The significance of these images is that the inequalities which produced them have roots in the theory of odd perfect numbers: