# Twin Prime Conjecture

**Type:** Conjecture  
**Domain:** Number Theory  
**Posed:** 1849, Alphonse de Polignac (general form)  
**Status:** OPEN (gap of exactly 2); bounded gaps (≤246) PROVEN  
**Codex URL:** /mathematics/number-theory/twin-prime-conjecture/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Infinitely many primes p where p+2 is also prime. Bounded-gap breakthrough 2013; exact gap of 2 remains open.

## Zhang's 2013 breakthrough
Yitang Zhang (previously outside mainstream academia) proved infinitely many prime pairs exist with SOME bounded gap (≤70,000,000) — first unconditional bounded-gap result ever.

## Polymath8 and Maynard-Tao
Collaborative online effort (Tao) reduced bound to 4,680; Maynard and (independently) Tao's new sieve method reduced it further to 246 (2014) — current record.

## Conditional result
Under the (unproven) Elliott-Halberstam conjecture, bound reduces to 6 — still not reaching 2.

## The parity problem
A fundamental, well-understood limitation of current sieve methods prevents them alone from reaching gap=2 without new mathematical ideas.

## Hardy-Littlewood k-tuple conjecture (1923)
Generalizes twin primes to arbitrary prime constellation patterns — also unproven, unifies with Goldbach-type problems.

## Sources
### Tier 1
- Zhang, Y. (2014). Annals of Mathematics.
- Maynard, J. (2015). Annals of Mathematics.
### Tier 2
- Goldston, Pintz and Yıldırım (2009). Annals of Mathematics.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*