Open problem, Arithmetic and number theory, posed 1849

Twin prime conjecture

OpenLandau #2

There are infinitely many primes $p$ such that $p+2$ is also prime.

Research frontier as of 2026

As of 2026 the twin prime conjecture is open. The unconditional record is $H_1 \le 246$ (Polymath8b, 2014), meaning some even gap of at most 246 recurs infinitely often among consecutive primes — but not necessarily the gap 2. Conditional on the generalized Elliott–Halberstam conjecture this drops to $H_1 \le 12$ , and to $H_1 \le 6$ under a further strengthened form, none of which are known unconditionally. All of these results rest on sieve methods that face the 'parity problem': sieves alone cannot distinguish numbers with an odd number of prime factors from those with an even number, which is why $H_1=2$ has never been reached by this route.

Best known results

Unconditional: $H_1 \le 246$ (Polymath8b, 2014), building on Zhang (2013, $70{,}000{,}000$ ) and Maynard–Tao's multidimensional sieve (2013).
Conditional on the generalized Elliott–Halberstam conjecture: $H_1 \le 12$ , and $H_1 \le 6$ under a further strengthened variant (Polymath8b, 2014).
The Maynard–Tao sieve also gives infinitely many bounded clusters of $m$ primes for every fixed $m$ , not only pairs.

Tools and where they stop

Tool	Achieved	Where it stops
GPY / Maynard–Tao multidimensional sieve	Gives an unconditional, explicit finite bound on the smallest gap that recurs infinitely often, currently $246$ .	Blocked by the sieve-theoretic parity problem from ever reaching gap $2$ without new structural input.
Distribution-of-primes conjectures (Elliott–Halberstam and generalizations)	If assumed, lower the conditional bound to $12$ , or $6$ under a strengthened form.	The conjectures themselves are unproved, so these bounds are conditional, not genuine progress toward an unconditional proof.
Collaborative optimization (Polymath project)	Squeezed the sieve weights and parameters to bring the unconditional bound from thousands down to $246$ within about a year of Zhang's paper.	Optimizing constants within the same method cannot, by itself, close the remaining gap from $246$ down to the exact value $2$ .

Open questions

Is there a way to break the sieve-theoretic parity problem that currently blocks unconditional bounds below roughly $246$ ?
Would a proof of the Elliott–Halberstam conjecture actually suffice to reach gap $2$ , or only bring it close (e.g. to $6$ )?

Proofs

Maynard's multidimensional sieve: bounded gaps between primes (2013)James Maynard, 2013Difficulty 5/5ResearchCondensed summary

References

Yitang Zhang (2014). Bounded gaps between primes
James Maynard (2015). Small gaps between primes · DOI:10.4007/annals.2015.181.1.7
D.H.J. Polymath (2014). Variants of the Selberg sieve, and bounded intervals containing many primes · DOI:10.1186/s40687-014-0012-7 · arXiv:1407.4897