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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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    @JacobSchlather,So,we don't know if it is possible...2011-12-08

1 Answers 1

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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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    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.2018-01-06