Math 2.0

Number theory Analytic number theory Theorem

Bounded gaps between primes

Builds on Prime numberLimit inferiorSelberg sieveBombieri–Vinogradov theorem

Statement

Let pnp_n denote the nn-th prime number. There is a constant HH such that pn+1−pn≤Hp_{n+1} - p_n \le H for infinitely many nn. Equivalently, lim inf⁡n→∞ (pn+1−pn)<∞,\liminf_{n \to \infty} \, (p_{n+1} - p_n) < \infty, where lim inf⁡\liminf is the limit inferior.

Remarks

Yitang Zhang proved the statement in 2013 with H=7⋅107H = 7 \cdot 10^7. James Maynard, and independently Terence Tao, gave a simpler proof with H=600H = 600, and the Polymath project lowered the bound to H=246H = 246. Maynard's method also shows that for every m≥1m \ge 1, lim inf⁡n→∞(pn+m−pn)<∞\liminf_{n \to \infty} (p_{n+m} - p_n) < \infty. The proof builds on the method of Goldston, Pintz and Yıldırım: a Selberg sieve with multidimensional weights detects a tuple n+h1,…,n+hkn + h_1, \ldots, n + h_k that contains at least two primes. The input on primes in arithmetic progressions is the Bombieri–Vinogradov theorem in Maynard's version and a strengthening of it for smooth moduli in Zhang's version. The twin prime conjecture is the claim that H=2H = 2 works; it remains open.

Going deeper

For a student

The statement is the first unconditional step towards the twin prime conjecture. Goldston, Pintz and Yıldırım had shown that bounded gaps would follow from a level of distribution of the primes greater than 1/21/2, which is beyond the Bombieri–Vinogradov theorem. Zhang proved such a level for a restricted class of moduli; Maynard changed the sieve weights so that the level 1/21/2 is already enough. Under the Elliott–Halberstam conjecture and its generalisations, the method gives smaller values of HH, down to 66.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptBig O notation, concept, no evidence levelBig O notationconceptChebyshev functions, concept, no evidence levelChebyshev functionsconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptDirichlet L-function, concept, no evidence levelDirichlet L-functionconceptDirichlet character, concept, no evidence levelDirichlet characterconceptDivisibility, definition, no evidence levelDivisibilitydefinitionEuler's totient function, concept, no evidence levelEuler's totient funct…conceptLarge sieve, concept, no evidence levelLarge sieveconceptLimit inferior, concept, no evidence levelLimit inferiorconceptPrime number, concept, no evidence levelPrime numberconceptSelberg sieve, concept, no evidence levelSelberg sieveconceptSiegel zero, concept, no evidence levelSiegel zeroconceptZero of a holomorphic function, concept, no evidence levelZero of a holomorphic…conceptvon Mangoldt function, concept, no evidence levelvon Mangoldt functionconceptPrime number, definition, no evidence levelPrime numberdefinitionGeneralized Riemann hypothesis, conjecture, no evidence levelGeneralized Riemann h…conjecturePrime-counting function, definition, no evidence levelPrime-counting functi…definitionPrime number theorem, statement, no evidence levelPrime number theoremstatementSiegel–Walfisz theorem, statement, no evidence levelSiegel–Walfisz theoremstatementBombieri–Vinogradov theorem, statement, no evidence levelBombieri–Vinogradov t…statementBounded gaps between primes, statement, no evidence levelBounded gaps between …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number field concept
  2. Big O notation concept
  3. Chebyshev functions concept
  4. Dedekind zeta function concept
  5. Dirichlet L-function concept
  6. Dirichlet character concept
  7. Divisibility definition
  8. Euler's totient function concept
  9. Large sieve concept
  10. Limit inferior concept
  11. Prime number concept
  12. Selberg sieve concept
  13. Siegel zero concept
  14. Zero of a holomorphic function concept
  15. von Mangoldt function concept
  16. Prime number definition
  17. Generalized Riemann hypothesis conjecture
  18. Prime-counting function definition
  19. Prime number theorem statement
  20. Siegel–Walfisz theorem statement
  21. Bombieri–Vinogradov theorem statement
  22. Bounded gaps between primes statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Prime_gap
About this page
TagXMXFDWYYMD
URIm2://XMXFDWYYMD
Kindstatement, theorem
WritingW0 stub
MSC11N05, 11N36
StatusThis copy of the site has no build data. Run m2 status XMXFDWYYMD.
AboutPrime number, Twin prime conjecture
UsesPrime number, Limit inferior, Selberg sieve, Bombieri–Vinogradov theorem
Changed2 November 2026

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