Catalan–Mersenne conjecture

The conjecture that repeatedly applying the Mersenne operation, starting from 2, produces only finitely many primes. The operation maps n to 2ⁿ−1.

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Mersenne number: An integer of the form 2ⁿ−1 for a positive integer n. Every step of the conjectured sequence applies this defining form to the preceding term.
Prime number: An integer greater than one whose only positive divisors are one and itself. The conjecture concerns how often primality occurs along the recursively defined sequence.
Marin Mersenne: A seventeenth-century French scholar whose name is attached to numbers of the form 2ⁿ−1. The conjecture iterates the operation associated with Mersenne numbers.
Infinitude of Mersenne primes: The open question of whether infinitely many primes have the form 2ᵖ−1. This asks about all Mersenne exponents, rather than the sparse exponents reached by iteration.
Mersenne prime: A prime number of the form 2ᵖ−1, where p is prime. The conjecture asks whether repeated Mersenne operations eventually stop producing primes.
Mersenne exponent: The exponent p in a Mersenne number 2ᵖ−1; a Mersenne number can be prime only if p is prime. Prime exponents are necessary, though not sufficient, for prime terms in this sequence.
Catalan's conjecture: The conjecture, proved by Preda Mihăilescu, that 8 and 9 are the only consecutive positive perfect powers. The shared name distinguishes this conjecture from a separate problem associated with Catalan.
Prime gaps: The differences between consecutive prime numbers, whose distribution remains incompletely understood. The conjecture concerns primes in an exceptionally sparse, rapidly growing sequence.
Fermat's little theorem: A theorem describing powers modulo a prime, central to elementary tests for primality. Modular arithmetic helps analyze divisibility among terms generated by the recurrence.
Modular arithmetic: Arithmetic in which numbers are identified by their remainders upon division by a fixed modulus. Congruences expose factors and restrictions on possible prime terms.