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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 ?

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

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

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  • $\begingroup$ thanks but my question is about known facts..not about conjectures.. $\endgroup$ – Peđa Terzić Dec 8 '11 at 8:28
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    $\begingroup$ The only known fact is that the number of Mersenne primes is somewhere between 40 and infinity. $\endgroup$ – Gerry Myerson Dec 8 '11 at 12:05
  • $\begingroup$ Look at "Mersenne prime" in Wikipedia.... Father Marin Mersenne was the scientific Internet of early 17th century Europe, in that he corresponded (by courier, before there were national postal services) with almost all the scientists of Europe.... So if you wanted something to be widely known, you told him. $\endgroup$ – DanielWainfleet Jan 6 '18 at 17:13

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