Are there infinite number of Mersenne Primes?

Definitions and mathematical statement

A Mersenne number is an integer of the form $2^n-1$, where $n\ge2$. When it is prime it is called a Mersenne prime. If $n=ab$ with $a,b>1$, then $2^a-1$ is a proper divisor of $2^n-1$, so a prime Mersenne number necessarily has a prime exponent. Consequently the infinitude question can be stated exactly as \[ \forall X>0\ \exists\text{ prime }p>X\quad 2^p-1\text{ is prime}. \] Equivalently the counting function $\#\{p\le x:p\text{ prime and }2^p-1\text{ prime}\}$ tends to infinity as $x\to\infty$. No upper bound on the gap between successful exponents is prescribed.

The primality of $p$ is necessary but not sufficient for the primality of $2^p-1$. The target concerns the latter integers themselves, not the number of prime divisors appearing among them. Either a proof of infinitely many successful exponents or a proof that all sufficiently large exponents fail would resolve the question. A finite collection of certified examples establishes only the existence of that many Mersenne primes, regardless of how large the examples are.

Short English statement

Are there infinitely many prime exponents p for which two to the pth power minus one is also prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3375 (OPG-59977), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Are there infinite number of Mersenne Primes?. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/are_there_infinite_number_of_mersenne_primes.