[#P28] Twin prime conjecture
Problem. There are infinitely many primes \(p\) for which \(p+2\) is also prime.
1Context
Bounded gaps between primes are known to occur infinitely often, while the specific gap 2 remains unresolved.
2Problem setup
Definition 1 (A prime). A prime is an integer greater than 1 whose only positive divisors are 1 and itself.
Definition 2 (A twin-prime pair consists of two primes whose difference). A twin-prime pair consists of two primes whose difference is 2.
Remark 1. Bounded gaps between primes are known to occur infinitely often, while the specific gap 2 remains unresolved.
3What counts as a solution
- Prove that infinitely many primes p have p + 2 prime, or prove that only finitely many such primes exist.
1Status
Current status (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.[2][1]
1Records
Notes and companion material
Original intake status. The cited American Mathematical Society publication identifies this conjecture as unsolved. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Bounded-gap theorems do not currently force the repeated gap to equal 2; consult current prime-gap literature before claiming novelty.
Recorded example 1. The pairs (3, 5), (5, 7), (11, 13), and (17, 19) are twin-prime pairs.
Computational notes
- Finding twin primes through any finite bound does not prove that infinitely many exist.
2See also
How to cite
TheoremDB contributors, “Twin prime conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/twin-prime-conjectureThis page as plain text: twin-prime-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from ams.org[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗website · primary source · checked 2026-07-31Source 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.Also cited at Section 1.9, Twin primes.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.Packet-linked problem statement.Source named by the research packet.
- Polymath8, Bounded gaps between primes, project wiki and final record table. michaelnielsen.org checked 2026-08-01. Current records table, m=1. ↗website · primary source · checked 2026-08-01Source use: original summary.Records the unconditional H_1<=246 theorem.
An original CC0 restatement prepared by TheoremDB maintainers.