Browse › Number theory › 026 Positive lower density of large prime gaps
Positive lower density of large prime gaps
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed $C\log p$, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which $p_n/n$ increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Terence Tao, Erdős Problem #968 forum thread (erdosproblems.com, comments of 8 and 10 Sep 2025). These comments predate the paper: they sketch a positive answer to the question in Corollary 1.2, conditional on RH and a weak pair-correlation hypothesis (via Heath-Brown), and call this "close to the limit of current technology".
Formalization
- OpenAI math release, Lean scope notes for family 026 (lean/docs/026.md). The release's scope note describes the formalized result as the prime-ratio corollary (positive lower density of indices with $p_n/n<p_{n+1}/(n+1)$) and links the comparator statement
ComparatorChallenges/PrimeGaps.lean. - OpenAI, lean/OAI/NumberTheory/PrimeGaps/Main.lean (openai/math, commit adc7f12). This file states
theorem main, a Lean version of Theorem 1.1 (for every $C>0$ there are $c>0$, $N_0$ with $c N\le$ #large-gap indices up to $N$), proved from a formalized Bombieri–Vinogradov theorem; the directory contains about 57 supporting files. - OpenAI, lean/OAI/NumberTheory/PrimeGaps/RatioCorollary.lean (openai/math, commit adc7f12). This file proves the ratio-increase density statement (Corollary 1.2) and the conjunction
large_gaps_and_ratio_density, usingLargePrimeGaps.main. - Google DeepMind formal-conjectures, FormalConjectures/ErdosProblems/968.lean (formal-conjectures). This file is an independent Lean statement of Erdős Problem 968 as
0 < {n | u n < u (n+1)}.lowerDensity; it is marked solved, with a docstring that credits the OpenAI preprint and links the OpenAI Lean proof, while the file itself keepssorry.
Related prior literature mentioned by others
- D. R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta-function (Acta Arith. 41 (1982), 85–99). Tao's 2025 erdosproblems comment uses an estimate from this paper for the conditional (RH plus pair correlation) answer to the question that Corollary 1.2 settles unconditionally; it is not in the paper's bibliography.
- Danilo Bazzanella, Alessandro Languasco and Alessandro Zaccagnini, Prime numbers in logarithmic intervals (arXiv:0809.2967, 2008; Transactions of the American Mathematical Society 362, 2010). The AI Proved What? exposition identifies this work, also cited in the manuscript, as the earlier unconditional positive-proportion result for thresholds below approximately 0.58 times the logarithmic gap scale.
- mo271, fix(ErdosProblems/968): the increasing and decreasing triple variants are true (formal-conjectures PR #6752, 30 Sep 2026). This PR settles Erdős's follow-up questions about increasing and decreasing triples of $p_n/n$ by citing Ford–Maynard–Tao, "Chains of large gaps between primes" (arXiv:1511.04468), and Maynard (2015); the paper does not cover these variants.
Possibly relevant literature (found by the agent, not checked)
- D. A. Goldston, J. Pintz, C. Y. Yıldırım, Positive Proportion of Small Gaps Between Consecutive Primes (arXiv:1103.3986, 2011). This paper proves the small-gap counterpart of Theorem 1.1: a positive proportion of consecutive gaps are below $\eta\log p_n$, counted with equal weight on prime indices. It is not in the bibliography.
- W. D. Banks, T. Freiberg, J. Maynard, On limit points of the sequence of normalized prime gaps (arXiv:1404.5094, Proc. LMS 2016). This paper studies the values taken by $d_n/\log p_n$ (its limit points) with Maynard-sieve methods, which bears on Theorem 1.1's count of gaps with $d_n/\log p_n>C$. It is not in the bibliography.
Digests and expositions
- AI Proved What?, Long gaps, how often? (Episode 026, October 2026). This video and transcript explain the claimed gap-frequency theorem, its prime-ratio consequence, and the adjacent-interval counting argument, with scripts and animations described as AI-assisted in the site's production notes. [AI-generated per source] [family-level]
News and blogs
- Thomas Bloom, Erdős problems in the OpenAI proofs (erdosproblems.com blog, 8 Oct 2026). The entry on Problem 968 summarizes the claimed answer, says the argument uses Goldston–Pintz–Yıldırım and Maynard–Tao sieve tools, and states that "This proof has been formalised."
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 026
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026,
author = {{OpenAI}},
title = {{Positive lower density of large prime gaps}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf}{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/026.md. The Lean statements have not been checked by people against the paper's claims.
Family 026: Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C\log p_n$, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $p_n/n$ increases have positive lower density, answering Erdős and Prachar.
This is the only paper in this family.