PartialNumber Theory·1849

Twin Prime Conjecture

Predicts infinitely many prime pairs separated by 2, such as 11 and 13.

The twin prime conjecture asserts that there are infinitely many primes p such that p + 2 is also prime. In 2013, Yitang Zhang stunned the mathematical world by proving that there are infinitely many pairs of primes differing by at most 70 million, the first finite bound ever established. James Maynard independently improved the bound later that year using a different sieve method, and the collaborative Polymath 8b project subsequently reduced the gap to 246, where it currently stands. The Hardy-Littlewood conjecture predicts the precise asymptotic density of twin primes via the twin prime constant C_2 = 0.6601...

Formula

Conjecture

\#\{p \le x : p\text{ and }p+2\text{ both prime}\} = \infty

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

Hardy-Littlewood asymptotic

\pi_2(x) \sim 2C_2 \int_2^x \frac{dt}{(\ln t)^2},\qquad C_2 = \prod_{p\ge 3}\left(1-\frac{1}{(p-1)^2}\right) \approx 0.6601

The first Hardy-Littlewood conjecture predicts the asymptotic count of twin primes using the twin prime constant.

Current best bound (Polymath 8b)

\liminf_{n\to\infty}(p_{n+1}-p_n) \le 246

There are infinitely many pairs of consecutive primes differing by at most 246. Gap 2 (the twin prime case) remains open.

Summary

The twin prime conjecture asserts that there are infinitely many primes p such that p + 2 is also prime. In 2013, Yitang Zhang stunned the mathematical world by proving that there are infinitely many pairs of primes differing by at most 70 million, the first finite bound ever established. James Maynard independently improved the bound later that year using a different sieve method, and the collaborative Polymath 8b project subsequently reduced the gap to 246, where it currently stands. The Hardy-Littlewood conjecture predicts the precise asymptotic density of twin primes via the twin prime constant C_2 = 0.6601...

Sources

Bounded gaps between primes

arXiv:1305.47262014

Small gaps between primes

Variants of the Selberg sieve, and bounded intervals containing many primes

arXiv:1407.48972014

Videos

Thumbnail for Twin Prime Conjecture - Numberphile

Twin Prime Conjecture - Numberphile

Thumbnail for Gaps between Primes - Numberphile

Gaps between Primes - Numberphile

Thumbnail for Twin Proofs for Twin Primes - Numberphile

Twin Proofs for Twin Primes - Numberphile

Researchers

YZ

Yitang Zhang

University of New Hampshire

JM

James Maynard

University of Oxford

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Alphonse de Polignac

France

Timeline

1849

Polignac proposes a broad prime gaps conjecture

He conjectures that for every even k, there are infinitely many consecutive prime pairs differing by k.

1915

Brun proves the sum of reciprocals of twin primes converges

Brun's theorem shows that even if twin primes are infinite, they are sparse enough for their reciprocal series to converge (Brun's constant ~ 1.902).

2013 Apr

Zhang proves bounded gaps between primes (gap < 70 million)

Yitang Zhang, a lecturer at the University of New Hampshire, establishes the first finite bound on prime gaps.

2013 Nov

Maynard independently improves the bound using a new sieve

James Maynard at Oxford obtains a simpler proof with a much smaller gap bound. He later receives the 2022 Fields Medal.

2014

Polymath 8b reduces the bound to 246

The collaborative Polymath project, combining Zhang's and Maynard's techniques, achieves the current best bound of H = 246.