# P35: Infinitely many Mersenne primes

- ID: `P35`
- Reference: `infinitely-many-mersenne-primes`
- Page: https://theoremdb.org/statements/P35
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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

### Problem setup

- **Definition (A Mersenne number).** A Mersenne number is a number of the form 2^n - 1.
- **Definition (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.** Mersenne primes include many of the largest explicitly known primes and are connected to even perfect numbers.

### What counts as a solution

- Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do.

## Status

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](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- The statement and status were checked against the cited AMS article and current records on 2026-07-22.
- Discovery of further individual Mersenne primes does not establish infinitude.

- Recorded example: The exponents 2, 3, 5, and 7 give the Mersenne primes 3, 7, 31, and 127.

### Open directions

- **Route 1** (reported): Prove that infinitely many prime exponents p give 2^p - 1 prime, or prove that only finitely many do. [1](#reference-1)

### Computational notes

- Distributed searches can discover new examples and eliminate bounded exponent ranges only.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `infinitely-many-mersenne-primes`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>American Mathematical Society Graduate Student Blog, The Infinitude of Mersenne Primes (2013), checked 2026-08-01. American Mathematical Society Graduate Student Blog, 2013 https://blogs.ams.org/mathgradblog/2013/07/18/infinitude-mersenne-primes/
   - Also cited at problem statement and unresolved infinitude discussion
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>Great Internet Mersenne Prime Search, List of Known Mersenne Prime Numbers, checked 2026-08-01. current known-primes table and provisional-ranking note https://www.mersenne.org/primes/
   - reference_database; primary source; checked 2026-08-01
   - Source use: original_summary
   - Provides the maintained finite frontier of 52 known examples without supporting infinitude.