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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
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.
