Proved
Unconditional H₁ ≤ 246 ↗
Published Polymath8b bound.
Observed Oct 13, 2026
Bound · version 2
Gaps at most 246, infinitely often
Statement
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 246; in other words .
Assumptions
- Consecutive positive primes.
- No additional conjectural hypothesis in the cited mathematical result.
Scope
An unconditional upper bound on the recurring gap, attributed to Polymath8b; it does not imply every gap is this small.
Explanation
In How small can prime gaps be? · publication v5
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| Gaps at most 246, infinitely often → Proved Result date not recorded | ≤ 246 | Unconditional bound for gaps between consecutive primes occurring infinitely often. |
In How small can prime gaps be? · publication v7
Bounds & computations · 1
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| Gaps at most 246, infinitely often → Proved Selected source date: | ≤ 246 | Unconditional bound for gaps between consecutive primes occurring infinitely often. |
Evidence
Source reports
Connections
In this problem · publication v5 · Selected context
How small can prime gaps be? →
Bound · Proved
Scope: The Polymath8b collaboration sharpened the bound in 2014.
Limit: The target H₁ = 2 is not established.
Nested results concern the same prime-gap limit; they are not separate portions of a solved problem.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
How small can prime gaps be? →
Bound · Proved
Scope: The Polymath8b collaboration sharpened the bound in 2014.
Limit: The target H₁ = 2 is not established.
Nested results concern the same prime-gap limit; they are not separate portions of a solved problem.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Variants of the Selberg sieve, and bounded intervals containing many primes →
- Main bounded-gap results and introduction → · Original passage ↗
Recorded wording
The published Polymath8b argument establishes an unconditional recurring prime gap at most 246; the twin-prime target remains out of reach.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
D. H. J. Polymath · Published · Publication date 2014
Peer-reviewed historical result.
Revision & attribution
Source revision: 2014. Observed Oct 13, 2026.
No AI contribution is asserted by this source record.
A new bound for small gaps between primes →
- Theorem 1.1; introduction’s prior bounds; Appendix A, Formal certificate by AxiomProver → · Original passage ↗
Recorded wording
The preliminary draft states H₁ ≤ 212 and explains the analytic and exact rational calculations. Its appendix reports a formal deduction with analytic and variational hypotheses.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
François Charton · Letong Hong · Kenny Lau · Ken Ono · Guillaume Remy · Ho Chung Siu · Ashvin A. Swaminathan · Jesse Thorner · Yunzhou Xie · Preprint · Publication date Sep 3, 2026
Public manuscript or exposition; publication does not imply independent OpenMath review.
Revision & attribution
Source revision: 2026-09-03. Observed Oct 13, 2026.
The nine authors supply the mathematics, building on Julia Stadlmann and earlier sieve work. AxiomProver is credited with generating a formal deduction; this is not a claim that the model independently discovered the mathematical bound.
Source updated Sep 3, 2026.
Bounded gaps between primes →
- Abstract → · Original passage ↗
Recorded wording
Zhang proves that lim inf (pₙ₊₁ − pₙ) < 7 × 10⁷, refining Goldston–Pintz–Yıldırım with a stronger Bombieri–Vinogradov estimate for moduli without large prime factors.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Yitang Zhang · Published · Publication date May 1, 2014
Annals of Mathematics 179(3), 1121–1174 (2014); DOI checked against Crossref.
Revision & attribution
Source revision: 2014-05-01. Observed Oct 21, 2026.
No AI contribution is asserted by this source record.
Small gaps between primes →
- Abstract → · Original passage ↗
Recorded wording
Maynard shows that lim inf (pₙ₊₁ − pₙ) ≤ 600, and ≤ 12 assuming the Elliott–Halberstam conjecture.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
James Maynard · Published · Publication date 2015
Annals of Mathematics 181, 383–413 (2015), DOI 10.4007/annals.2015.181.1.7 checked against Crossref; arXiv v3 pinned by content hash.
Revision & attribution
Source revision: 2015. Observed Oct 21, 2026.
No AI contribution is asserted by this source record.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Unconditional H₁ ≤ 246 ↗
Published Polymath8b bound.
Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-prime-gaps-bound-246
- Statement version
- 2
- SHA-256
- 014283da3280bdffc86b2276ef913027658f22276d2617a580c148e6746d0eb4