Are there infinitely many Mersenne primes?

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 ?

• Do you know about the LPW conjecture? – J. M. is a poor mathematician Dec 8 '11 at 8:18
• @J.M.,Interesting,but it isn't fact,it is conjecture... – Peđa Terzić Dec 8 '11 at 8:21
• As far as I know, this is still an open problem. – InterestedGuest Dec 8 '11 at 8:23
• Clearly, you missed the point. There's a reason why the infinitude of Mersenne primes remains a conjecture. – J. M. is a poor mathematician Dec 8 '11 at 8:30
• No, he's saying we don't know. – JSchlather Dec 8 '11 at 8:41