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Infinitude of Mersenne primes

Determine whether there are infinitely many Mersenne primes, namely primes of the form 2^p − 1 with p itself prime.

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Background

The paper develops generalized polynomial frameworks and an enhanced Lucas–Lehmer test aimed at studying Mersenne numbers and primes. Despite these advances, the classical question of whether Mersenne primes are infinite in number is highlighted as unresolved. The authors emphasize this as an enduring challenge in number theory and a motivation for their broader program connecting Ψ-polynomials to Mersenne arithmetic.

References

The tantalizing question of whether there exists an infinite number of Mersenne primes remains an enigma in mathematics. The finite or infinite status of the known Mersenne primes continues to elude us, representing an ongoing field of active investigation (as cited in [1, 4, 8, 11, 13, 21]).

— Generalizing the Eight Levels Theorem: A Journey to Mersenne Prime Discoveries and New Polynomial Classes (2404.05772 - Ibrahim, 8 Apr 2024) in Introduction (Lucas-Lehmer Primality Test paragraph)