[#P35] Infinitely many Mersenne primes
Problem. Infinitely many exponents \(p\in\mathbb{P}\) have the property that \(2^p-1\in\mathbb{P}\), where \(\mathbb{P}\) is the set of prime numbers.
1Context
Mersenne primes include many of the largest explicitly known primes and are connected to even perfect numbers.
2Problem setup
Definition 1 (A Mersenne number). A Mersenne number is a number of the form 2^n - 1.
Definition 2 (If 2^n - 1). If 2^n - 1 is prime, then n must be prime, though a prime exponent does not guarantee a prime value.
Remark 1. Mersenne primes include many of the largest explicitly known primes and are connected to even perfect numbers.
3What counts as a solution
- Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: GIMPS lists 52 known Mersenne primes, with the latest M_136279841 and provisional ordering because an interval has not been exhausted. This expanding finite list does not prove infinitude. Exact unresolved remainder: Prove that infinitely many prime exponents p make 2^p-1 prime, or prove that only finitely many do.[2][1]
1Records
Notes and companion material
Original intake status. The cited AMS article identifies the infinitude of Mersenne primes as unresolved, and current public status was checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Discovery of further individual Mersenne primes does not establish infinitude.
Recorded example 1. The exponents 2, 3, 5, and 7 give the Mersenne primes 3, 7, 31, and 127.
Computational notes
- Distributed searches can discover new examples and eliminate bounded exponent ranges only.
2See also
How to cite
TheoremDB contributors, “Infinitely many Mersenne primes,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/infinitely-many-mersenne-primesThis page as plain text: infinitely-many-mersenne-primes.md
This problem includes 2 records joined by 2 typed links, sourced from blogs.ams.org[1], current as of July 31, 2026.
1References
- Packet source. American Mathematical Society Graduate Student Blog, The Infinitude of Mersenne Primes (2013), checked 2026-08-01. American Mathematical Society Graduate Student Blog, 2013. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited AMS article identifies the infinitude of Mersenne primes as unresolved, and current public status was checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at problem statement and unresolved infinitude discussion.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the infinitude question and its mathematical context.Source named by the research packet.
- Great Internet Mersenne Prime Search, List of Known Mersenne Prime Numbers, checked 2026-08-01. current known-primes table and provisional-ranking note. ↗reference database · primary source · checked 2026-08-01Source use: original summary.Provides the maintained finite frontier of 52 known examples without supporting infinitude.
An original CC0 restatement prepared by TheoremDB maintainers.