Polignac's conjecture

Statement

For every natural number $n$, there exist infinitely many prime numbers $p$ such that the prime gap between $p$ and the next prime is exactly $2n$.

In particular, it asserts that there are infinitely many twin primes, infinitely many cousin primes, and infinitely many sexy primes.

Relation with other facts/conjectures

Stronger conjectures

• Hardy-Littlewood conjecture

Weaker facts and conjectures

• Twin primes conjecture
• Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam