[#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.

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Definitions and notation

1Context

Mersenne primes include many of the largest explicitly known primes and are connected to even perfect numbers.

2Definitions

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.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Infinitely many Mersenne primes, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do.^[1] Relevance to this problem For Infinitely many Mersenne primes, record infinitely-many-mersenne-primes-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Infinitely many Mersenne primes.” TheoremDB. P35. Problem statement; statement text SHA-256 9bcb6424546b406ed443abd280d0bcd1f6066a4cd5e59c6aaa9b8024d5bb4375. https://theoremdb.org/statement/?ref=P35

BibTeX

@misc{theoremdb-problem-9bcb6424546b406ed443abd280d0bcd1f6066a4cd5e59c6aaa9b8024d5bb4375,
  title = {{Infinitely many Mersenne primes}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 9bcb6424546b406ed443abd280d0bcd1f6066a4cd5e59c6aaa9b8024d5bb4375},
  url = {https://theoremdb.org/statement/?ref=P35}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from blogs.ams.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P35] Infinitely many Mersenne primes

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