Proved
Zhang: H₁ < 7 × 10⁷ ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Bounded gaps between primes · Abstract →
Observed Oct 21, 2026
Bound · version 3
Gaps at most 70,000,000, infinitely often
Statement
For every real X, there are consecutive primes p < q with p ≥ X and q − p ≤ 70000000; 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 Zhang; it does not imply every gap is this small.
Explanation
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 70,000,000, infinitely often → Proved Selected source date: | ≤ 70,000,000 | Unconditional bound for gaps between consecutive primes occurring infinitely often. |
Evidence
Source reports
Proved
Historical bounds recorded in the paper ↗
Zhang, Maynard and Stadlmann bounded-gap results.
Observed Oct 13, 2026
Replaces an earlier source assessment. Version history ↓
Connections
In this problem · publication v7 · Selected context
How small can prime gaps be? →
Bound · Proved
Scope: Zhang established the first finite recurring-gap bound in 2013.
Limit: The target H₁ = 2 is not established.
No overlap is counted as an additional resolved part.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
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.
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 · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Zhang: H₁ < 7 × 10⁷ ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →
Proved
Historical bounds recorded in the paper ↗
Zhang, Maynard and Stadlmann bounded-gap results.
Observed Oct 13, 2026 · Source passage →
Replaces an earlier source assessment; see the original source for its history.
Record details
- Claim ID
- claim-prime-gaps-bound-70000000
- Statement version
- 3
- SHA-256
- 22fbc6f61d2b32dc15a282dd9eed75880a1d1d2ef9a7537cf466b6a8b406cf21