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The joint Dickman law for consecutive integers

OpenAI · September 24, 2026

Lean

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Abstract

Let $P^+(n)$ denote the largest prime factor of n. We prove that $\log P^+(n)/\log n$ and $\log P^+(n+1)/\log n$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering $P^+(n)\lt P^+(n+1)$ has natural density 1/2.

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Notes

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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.

Follow-up works

  • Ji Ho Bae (jbaelaw), Erdos928 (GitHub repository, Lean 4/Mathlib, accessed 2026-10-21). Formalizes in Lean a counting estimate and an explicit implication from the joint Dickman law of this paper to Erdős Problem 928; it explicitly does not re-prove or import the paper's analytic theorem. [AI-generated per source]

Comments and discussion

  • Thomas Bloom, Erdős problems in the OpenAI proofs (erdosproblems.com blog, 8 Oct 2026). Lists Problem 371 among the release's claimed results, notes that the 84-page PDF proves it as a corollary of a more general joint distribution law and that a Lean formalisation exists, and says he had not checked the proofs or formalisations himself. [family-level]

Formalization

Possibly relevant literature (found by the agent, not checked)

  • Zhiyuan Yang, Note on the largest prime factors of consecutive integers (arXiv:2610.23280, 16 Oct 2026). Gives an alternative proof of the Tao–Teräväinen positive-proportion results on patterns of $P^+$ at consecutive integers, discussed in the paper's "Previous work" section, and a lower bound $\gg x/\log x$ for $P^+(n+1)/P^+(n)\in[1,1+\varepsilon)$; it does not cite the release.

Digests and expositions

  • eigendash, openai-math-visualised (GitHub repository, accessed 2026-10-21). Includes a numerical probe of family 012 up to $6\times10^7$: the density of $P^+(n)<P^+(n+1)$ is reported as 0.499983, with a comparison of the binned joint distribution against the product of Dickman marginals.

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-joint-Dickman-law-for-consecutive-integers-September-24-2026,
  author = {{OpenAI}},
  title = {{The joint Dickman law for consecutive integers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf}{OAI:The-joint-Dickman-law-for-consecutive-integers-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/012.md. The Lean statements have not been checked by people against the paper's claims.

Family 012: Independent largest prime factors of consecutive integers

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^+(n)\lt P^+(n+1)$ have density 1/2.

This is the only paper in this family.