# P28: Twin prime conjecture

- ID: `P28`
- Reference: `twin-prime-conjecture`
- Page: https://theoremdb.org/statements/P28
- Record maturity: Reviewed problem with recorded work

## Problem

There are infinitely many primes \(p\) for which \(p+2\) is also prime.

### Context

Bounded gaps between primes are known to occur infinitely often, while the specific gap 2 remains unresolved.

### Problem setup

- **Definition (A prime).** A prime is an integer greater than 1 whose only positive divisors are 1 and itself.
- **Definition (A twin-prime pair consists of two primes whose difference).** A twin-prime pair consists of two primes whose difference is 2.
- **Remark.** Bounded gaps between primes are known to occur infinitely often, while the specific gap 2 remains unresolved.

### What counts as a solution

- Prove that infinitely many primes p have p + 2 prime, or prove that only finitely many such primes exist.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Polymath8b proves infinitely many intervals of length 246 contain two primes. This does not show any fixed gap, including 2, occurs infinitely often. Exact unresolved remainder: Prove infinitely many primes p have p+2 prime, or prove 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: Polymath8b proves infinitely many intervals of length 246 contain two primes. This does not show any fixed gap, including 2, occurs infinitely often. Exact unresolved remainder: Prove infinitely many primes p have p+2 prime, or prove only finitely many do.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Polymath8b proves infinitely many intervals of length 246 contain two primes. This does not show any fixed gap, including 2, occurs infinitely often.

Exact unresolved remainder: Prove infinitely many primes p have p+2 prime, or prove only finitely many do.

### Background and intake notes

- Original intake status: The cited American Mathematical Society publication identifies this conjecture as unsolved. The source and public status were 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 American Mathematical Society publication on 2026-07-22.
- Bounded-gap theorems do not currently force the repeated gap to equal 2; consult current prime-gap literature before claiming novelty.

- Recorded example: The pairs (3, 5), (5, 7), (11, 13), and (17, 19) are twin-prime pairs.

### Open directions

- **Route 1** (reported): Prove that infinitely many primes p have p + 2 prime, or prove that only finitely many such primes exist. [1](#reference-1)

### Computational notes

- Finding twin primes through any finite bound does not prove that infinitely many exist.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `twin-prime-conjecture`, 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>Evens, Odds, and Primes: A Taste of Number Theory, source checked for the TheoremDB status review (2026-07-31). American Mathematical Society book preview, section 1.9, Twin primes: An excursion into the unknown https://www.ams.org/bookstore/pspdf/amstext-61-prev.pdf
   - Also cited at Section 1.9, Twin primes
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The cited American Mathematical Society publication identifies this conjecture as unsolved. The source and public status were 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.
   - Packet-linked problem statement.
   - Source named by the research packet.
2. <a id="reference-2"></a>Polymath8, Bounded gaps between primes, project wiki and final record table. michaelnielsen.org checked 2026-08-01. Current records table, m=1 https://michaelnielsen.org/polymath/index.php?title=Bounded_gaps_between_primes
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Records the unconditional H_1<=246 theorem.