TheoremProved

Bounded gaps between primes (Zhang, Maynard)

Statement

$\displaystyle \liminf_{n\to\infty} (p_{n+1}-p_n) < \infty$ : there is a finite bound $C$ such that infinitely many pairs of consecutive primes differ by at most $C$ . The best known unconditional bound, from the Polymath8 project, is $C=246$ .

Why is it true?

Even though the average gap between primes near $x$ grows like $\ln x$ , this theorem shows the gaps do not grow without bound everywhere — infinitely often, two consecutive primes stay within a fixed bounded distance of each other, a weak form of the still-open twin prime conjecture ( $C=2$ ).

Proof sketch

Zhang (2013) adapted the Goldston–Pintz–Yıldırım sieve, combined with a new equidistribution estimate for primes in arithmetic progressions to moduli slightly beyond what the Bombieri–Vinogradov theorem gives unconditionally, to show $C\le 70{,}000{,}000$ . Maynard, and independently Tao, then introduced a more flexible multidimensional sieve (optimizing over many linear forms at once), simplifying the argument and lowering the bound to $C\le600$ . The Polymath8 collaborative project combined and further optimized these techniques, reaching $C\le246$ unconditionally (and $C\le6$ under the generalized Elliott–Halberstam conjecture).

Proved by

proved 2013Yitang Zhang proved 2013James Maynard

Topics that use this theorem

Prime numbers

Related theorems

Infinitude of primes (Euclid)Ternary Goldbach theorem (Helfgott)Green–Tao theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Yitang Zhang (2014). Bounded gaps between primes · DOI:10.4007/annals.2014.179.3.7
James Maynard (2015). Small gaps between primes · DOI:10.4007/annals.2015.181.1.7 · arXiv:1311.4600