# Bounded gaps between primes

- kind: problem
- status: solved (evidence tier V3)
- year: 1849

**Statement.** Some fixed gap between consecutive primes occurs infinitely often.

Zhang proved that some gap below 70 million recurs infinitely often, the first finite bound ever. Within a year, the Polymath8 project and Maynard's simpler multidimensional sieve brought the bound down to 246. It is one of the clearest recent examples of a public collaboration improving a result in the open. Under the Elliott–Halberstam conjecture the bound drops to 6.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed c. Implicit in the twin prime question. Closed 2013, Yitang Zhang; bound improved by James Maynard and Polymath8.

## Sources

- [Wikipedia: Prime gap](https://en.wikipedia.org/wiki/Prime_gap)
- [Wikipedia: Polymath Project](https://en.wikipedia.org/wiki/Polymath_Project)

[Number theory](/wiki/Space:number-theory.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- used by [twin-prime-conjecture](/wiki/Problem:twin-prime-conjecture.md) — Progress here drew on Bounded gaps between primes, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

---
Revision 5, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:bounded-prime-gaps?action=history. Edit: POST /api/wiki/page/Problem:bounded-prime-gaps {source, summary, base_rev: 5}.