Infinitude of Mersenne Primes

The conjecture asserts that $2^p-1$ is prime for infinitely many primes $p$. Such primes are named for Marin Mersenne, whose 1644 list of exponents purported to identify the prime values in a finite range [Mersenne1644]; the list was partly erroneous, and since no clearly documented statement of the modern infinitude conjecture survives, the question carries no reliable date of origin.

It is elementary that $2^m-1$ can be prime only when $m$ itself is prime, which thins the search but decides nothing. Wagstaff's study of the divisors of Mersenne numbers supports heuristics under which prime values should continue to appear indefinitely, at a slowly growing rate [Wagstaff1983Mersenne], and the distributed GIMPS search has tested enormous exponents and found many examples, including the largest primes currently known [GIMPSStatus].

The sequence $2^p-1$ grows exponentially, and no known method proves the infinitude of primes in so sparse a sequence; indeed it is not even known that infinitely many of the numbers $2^p-1$ with $p$ prime are composite. The conjecture remains open, and a resolution would apparently require fundamentally new tools.