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 ?