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

Theorem 1.1: $h(k)\le Ck^2/(\log\log(3k))^2$ for an absolute constant $C$ and every $k\ge1$ , uniformly over prime sets and interval positions; in particular $h(k)\ll k^2$ , answering Jacobsthal's question and improving Iwaniec's $k^2\log^2k$ . 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 $h(k)$ (Erdos #970's broader ask) is not determined: the gap to the lower bound near $k(\log k)^2$ remains. Theorem 1.2 of the same paper also bounds the quantity in Erdős Problem #687: every choice of one class per prime $p\le z$ leaves survivors below $z^2/(\log z)^2$ , so $Y(z)<z^2/(\log z)^2$ ; the connection was pointed out in a VibeMathed submission on 14 October 2026.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Jacobsthal's quadratic-bound question: is h(k) = O(k^2)? · Jacobsthal quadratic bound · Problem 970

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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

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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.