Twin prime conjecture

This article states a conjecture about there existing infinitely many of the following numbers/structures: twin primes
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Statement

There are infinitely many twin primes. In other words, there are infinitely many odd primes $p$ such that $p+2$ is also a prime.

In other words, the limit inferior of all prime gaps is $2$.

Relation with other conjectures and known facts

Infimum of prime gaps

The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is $2$. There are various results that prove bounds on the limit inferior:

• Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam: The Elliott-Halberstam conjecture, by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most $16$.
• Goldston-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.

Average prime gap

The prime number theorem states that the average prime gap is the natural logarithm of the prime.

Supremum of prime gaps

• Bertrand's postulate (which is in fact a theorem) states that there always exists a prime between any number and its double.

Generalizations

• Schinzel's hypothesis H is a much stronger and more general conjecture that provides a framework within which the twin primes conjecture fits.