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Jacobsthal's quadratic-bound question: is h(k) = O(k^2)?

Erdős problem #970 · erdosproblems.com/970

Let h(k)h(k) be the least mm such that every interval of mm consecutive integers contains an integer coprime to any prescribed positive integer with at most kk distinct prime factors. Brun's sieve gives h(k)≪kCh(k)\ll k^{C}; Iwaniec's linear-sieve work gave h(k)≪k2log⁡2kh(k)\ll k^2\log^2k (1978), improving Vaughan's k2log⁡4kk^2\log^4k; the best lower bound, from Ford, Green, Konyagin, Maynard and Tao, is about k(log⁡k)2log⁡log⁡log⁡k/log⁡log⁡kk(\log k)^2\log\log\log k/\log\log k. Jacobsthal asked, and Erdos recorded (1962; Erdos Problem #970), whether the upper bound is quadratic. Is h(k)≪k2h(k)\ll k^2?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Sieve theory: Jacobsthal's function
Posed by
E. Jacobsthal (papers from 1960); recorded by P. Erdos (1962, p. 163) and as Erdos Problem #970
Year posed
1962
Years open
64y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
34 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: h(k)≤Ck2/(log⁡log⁡(3k))2h(k)\le Ck^2/(\log\log(3k))^2 for an absolute constant CC and every k≥1k\ge1, uniformly over prime sets and interval positions; in particular h(k)≪k2h(k)\ll k^2, answering Jacobsthal's question and improving Iwaniec's k2log⁡2kk^2\log^2k. The proof pushes the linear sieve to its limiting parameter, keeping a boundary term, and controls the discrepancy with the actual residue classes through an inverse estimate and stopped counts. The constant is not explicit, and the true order of h(k)h(k) (Erdos #970's broader ask) is not determined: the gap to the lower bound near k(log⁡k)2k(\log k)^2 remains.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Jacobsthal's question as recorded by Erdos and on erdosproblems.com; the proof was not refereed. Lean: formalization.yaml lists OAI.Erdos970.Erdos970Final.erdos_970_quadratic (Comparator challenge Jacobsthal, file OAI/NumberTheory/Jacobsthal/Conclusions/QuadraticBound.lean). Its statement was read here: there is C > 0 such that for every k >= 1 some m <= C k^2 has the property that, for every positive n with at most k distinct prime factors and every integer start a, one of a, a+1, ..., a+m-1 is coprime to n. That is exactly Jacobsthal's question. The separate challenge JacobsthalImproved (erdos_970_iterated_log, same solution module, not in the formalization catalogue) states the paper's stronger bound C k^2/(log log 3k)^2. Not rebuilt here.

Sources

Changelog1 change

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