Infinitude of Mersenne primes

Determine whether there are infinitely many primes of the form 2^n − 1 (Mersenne primes), equivalently whether Euclid’s construction yields infinitely many even perfect numbers of the form (2^n − 1)·2^{n−1} with 2^n − 1 prime.

Background

Euclid’s Proposition IX.36 shows that when 2^n − 1 is prime, the number (2^n − 1)·2^{n−1} is perfect. Leonhard Euler later proved that all even perfect numbers arise this way, linking the infinitude of even perfect numbers directly to the infinitude of Mersenne primes.

Despite extensive computational efforts (e.g., GIMPS) and theoretical advances, it remains unresolved whether there are infinitely many primes of the form 2^n − 1, and thus whether Euclid’s family contains infinitely many members.