Result 021, Number theory

A quadratic bound for Jacobsthal’s function

Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

Lean formalization New or sharp bound

The bigger picture

Why it matters

How long can a stretch of integers avoid every number coprime to a fixed integer? The manuscript claims a bound controlled only by the number of distinct prime divisors, not by how large those primes are.

What changes?

Jacobsthal's function h(k) is the shortest interval length guaranteeing an integer coprime to any prescribed positive integer with at most k distinct prime divisors. Coprime means sharing no prime divisor. The unreviewed manuscript reports that, for positive k, h(k) is at most an absolute constant times k squared divided by the square of log log(3k). This is stronger than a quadratic bound, removes the classical logarithmic loss, and holds uniformly over prime choices and interval positions.

What does that help mathematicians do?

The claim limits how effectively k primes can collectively cover consecutive integers by divisibility: they cannot sustain a stretch without a coprime integer beyond the stated bound. Partitioning a long interval into blocks of the guaranteed length also yields a lower bound on its number of coprime integers. Researchers could use this guarantee without estimating the sizes or spacing of the prescribed prime divisors.

Are there practical applications?

The immediate value is foundational in number theory. The bound sharpens a guarantee for finding integers that simultaneously avoid divisibility by a specified collection of primes, a basic constraint in arithmetic problems. The supplied sources establish no practical search algorithm or runtime improvement; the reported advance concerns how long an interval must be to guarantee existence.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A quadratic bound for Jacobsthal's function

September 25, 2026 83 pages Main result formalized in Lean

Let $h(k)$ be the least integer such that every interval of $h(k)$ consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove $h(k)\ll k^2/(\log\log(3k))^2$ , giving an affirmative answer to Jacobsthal's quadratic-bound question.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026,
  author = {{OpenAI}},
  title = {{A quadratic bound for Jacobsthal's function}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026/paper.pdf}{OAI:A-quadratic-bound-for-Jacobsthals-function-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/021.md.

A quadratic bound for Jacobsthal’s function

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

A quadratic bound for Jacobsthal's function

Scope

Let $h(k)$ be the least interval length that guarantees an integer coprime to any prescribed positive modulus with at most $k$ distinct prime factors. The formalization proves the paper's strengthened Jacobsthal bound $h(k)\le Ck^2/(\log\log(3k))^2$ for one absolute $C>0$ and every $k\ge1$ . Intervals may begin at any signed integer. The earlier quadratic bound is also retained as a separate statement; the optimal order of growth is not determined.

Comparator links

Result	Comparator statement
Quadratic Jacobsthal bound	Jacobsthal.lean
Jacobsthal bound with an iterated-logarithm saving	JacobsthalImproved.lean

At a glance

Subject: Number theory
Manuscripts: 1
Pages: 83
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Jacobsthal's quadratic-bound questionProved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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