# Twin prime conjecture

- kind: problem
- status: open
- year: 1849
- authors: Alphonse de Polignac (general form)

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

Primes thin out, but they seem to keep arriving in close pairs forever. Sieve methods can count almost-primes but run into the parity barrier, which stops them from telling a number with an even count of prime factors from one with an odd count. The 2013 breakthrough on bounded gaps got within a constant of the conjecture: some gap of at most 246 occurs infinitely often. Closing 246 down to 2 appears to need an idea that breaks parity.

Posed 1849, Alphonse de Polignac (general form).

## What would settle it

Lean statement in formal-conjectures

## Related problems

- Progress here drew on [Bounded gaps between primes](/wiki/Problem:bounded-prime-gaps.md), whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Sources

- [Wikipedia: Twin prime](https://en.wikipedia.org/wiki/Twin_prime)

[Number theory](/wiki/Space:number-theory.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- uses [bounded-prime-gaps](/wiki/Problem:bounded-prime-gaps.md) — Progress here drew on Bounded gaps between primes, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

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