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There are 3 known Twin Mersenne Primes: $M3$ and $M5$, $M5$ and $M7$, $M17$ and $M19$. More precisely, if both $M(p)$ and $M(p+2)$ are both prime, then they are called Twin Mersenne Primes.

My question is: Is there is any result claiming that there cannot be any other twin Mersenne primes

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    $\begingroup$ There is no result on whether or not there are infinitely many Mersenne primes, so I would doubt that there is any result on whether or not there are infinitely many twin Mersenne primes. To make this case stronger, there is no result even on whether or not there are infinitely many twin primes. $\endgroup$ Dec 6, 2014 at 8:59
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    $\begingroup$ @barak, OP is not asking whether there are infinitely many, but whether there are any (beyond M19). $\endgroup$ Dec 6, 2014 at 9:09
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    $\begingroup$ @GerryMyerson: True. If there was a large amount of known Mersenne primes (even without a formal proof for an infinite amount of them), then my case would be very weak indeed. But since there are only some $40$ known Mersenne primes, asking whether or not there are any Mersenne twin primes beyond the $7$th Mersenne prime is "rather equivalent" to asking whether or not there is a finite amount of Mersenne twin primes out of the (yet unproven) infinite amount of Mersenne primes. I realize that it's not a "very mathematical" statement, so I hope you can read my point in between the lines. $\endgroup$ Dec 6, 2014 at 9:22

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p = 3, 5, 17, 107 and 521 are the only numbers known where $2^p - 1$ is a Mersenne prime and p+2 is a prime. Since Mersenne primes are so rare, Mersenne primes where p is a twin prime are even rarer.

I don't know of any particular reason why p+2 shouldn't be a prime and $2^{p+2} - 1$ shouldn't be a Mersenne prime if $2^p - 1$ is a Mersenne prime, so I wouldn't bet my life that there are no other twin Mersenne primes.

On the other hand, the primes p where $2^p - 1$ is a Mersenne prime seem to have an exponential distribution (very roughly the same number with x ≤ p < 10x for every x), so I wouldn't think the chance is more than one in 100,000 that another p exists. The last two known Mersenne primes have p ≈ 58,000,000 and q ≈ 74,000,000, so the chances that p+2 belongs to a Mersenne prime were quite slim.

You could ask whether there are consecutive primes p, q such that $2^p - 1$ and $2^q - 1$ are both Mersenne primes; there are two more examples with p = 2 and p = 13, but still not very likely.

You could ask whether there are more p with $2^p - 1$ a Mersenne prime and p+2 a prime; heuristically there should be infinitely many but they should be so rare that none might ever be found.

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