8
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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 ?

  • 1
    Do you know about the [LPW conjecture](https://en.wikipedia.org/wiki/Mersenne_conjectures#Lenstra.E2.80.93Pomerance.E2.80.93Wagstaff_conjecture)?2011-12-08
  • 0
    @J.M.,Interesting,but it isn't fact,it is conjecture...2011-12-08
  • 1
    As far as I know, this is still an open problem.2011-12-08
  • 0
    Clearly, you missed the point. There's a reason why the infinitude of Mersenne primes remains a **conjecture**.2011-12-08
  • 0
    J.M.,So you are saying that it is impossible to prove or disprove that there are infinitely many Mersenne primes ?2011-12-08
  • 2
    No, he's saying we don't know.2011-12-08
  • 0
    @JacobSchlather,So,we don't know if it is possible...2011-12-08

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