[Math] Are there infinitely many Mersenne primes

Question

known facts :

$1.$ There are infinitely many Mersenne numbers : $M_p=2^p-1$

$2.$ Every Mersenne number greater than $7$ is of the form : $6k\cdot p +1$ , where $k$ is an odd number

$3.$ There are infinitely many prime numbers of the form $6n+1$ , where $n$ is an odd number

$4.$ If $p$ is prime number of the form $4k+3$ and if $2p+1$ is prime number then $M_p$ is composite

What else one can include in this list above in order to prove (or disprove) that there are infinitely many Mersenne primes ?

Answer 1

It is not known whether or not there are infinitely many Mersenne primes.

Look at Mersenne conjectures, especially Lenstra–Pomerance–Wagstaff conjecture.