Result 012, Number theory

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+1n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P+(n)<P+(n+1)P^+(n)\lt P^+(n+1) have density 1/2.

Lean formalization Proof

The bigger picture

Why it matters

Consecutive integers sit next to each other, but how closely are their largest prime factors linked? The manuscript reports that, when measured on a logarithmic scale, these factors become statistically independent in the large-number limit.

What changes?

The largest prime factor is the biggest prime dividing an integer. The manuscript studies the logarithms of these factors for n and n+1, dividing both by the logarithm of n. It claims their joint limiting distribution is the product of two Dickman distributions, the individual laws describing these scaled sizes. This is independence in ordinary natural density: each integer up to a growing cutoff receives equal weight. The claim concerns this limiting distribution, not exact independence at finite cutoffs.

What does that help mathematicians do?

For fixed thresholds, researchers can therefore deduce the limiting proportion of consecutive pairs whose largest prime factors lie below the corresponding powers of n: it is the product of the two individual Dickman probabilities. The manuscript also reports that the first integer has the smaller largest prime factor with density one-half. This gives a precise statistical balance despite the arithmetic constraint of being consecutive.

Are there practical applications?

The immediate value is foundational, rather than a demonstrated practical application. The claimed law supplies a joint statistical description of prime factorization for neighboring integers, allowing number theorists to replace an independence heuristic with precise limiting proportions in this setting. The supplied material gives no convergence rate or computational performance claim.

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

The joint Dickman law for consecutive integers

September 24, 2026 84 pages

Let P+(n)P^+(n) denote the largest prime factor of n. We prove that log⁡P+(n)/log⁡n\log P^+(n)/\log n and log⁡P+(n+1)/log⁡n\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)<P+(n+1)P^+(n)\lt P^+(n+1) has natural density 1/2.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/012.md.

Independent largest prime factors of consecutive integers

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

Scope

Let P+(n)P^+(n) be the largest prime factor of nn. The formalization proves the joint Dickman law in ordinary natural density: for every 0<a,b<10<a,b<1, the density of integers satisfying P+(n)≤naP^+(n)\le n^a and P+(n+1)≤nbP^+(n+1)\le n^b tends to ρ(1/a)ρ(1/b)\rho(1/a)\rho(1/b), where ρ\rho is the Dickman function.

It also proves that each of the orderings P+(n)<P+(n+1)P^+(n)<P^+(n+1) and P+(n+1)<P+(n)P^+(n+1)<P^+(n) has natural density 1/21/2.

Comparator links

Result Comparator statement
Joint Dickman law and equal ordering densities for consecutive integers JointDickman.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.