oai-math-revised beta

Browse › Number theory › 011 Prime-factor statistics of $p-1$

The Poisson-Dirichlet law for prime predecessors

OpenAI · September 24, 2026

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

For a prime p chosen uniformly from $3\le p\le x$, list the prime factors of $p-1$ in decreasing order, with multiplicity. As $x\to\infty$, their logarithms, divided by $\log(p-1)$, converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

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

Formalization

  • Plasma AI, erdos-408 (Lean 4) (Palomar PALOMAR-2026-10-19-000009 v1, 2026-10-19). The README says the extensions of Theorems 3.1 and 6.1, the block sieve (Lemma 2.6) and Propositions 6.2–6.3 are proved in Lean, with graph inputs from vendored totient-fibers-lean4 code. [family-level]
  • Yao Xu (@cadamcat), artin-primitive-roots-lean4 (GitHub, accessed 2026-10-21). A Lean 4 formalization of an Artin primitive-root result from the release whose README names this paper's Theorem 3.1 as the marked prime estimate used in part 1, without saying whether that estimate is itself formalized. [AI-generated per source]

Digests and expositions

  • eigendash, openai-math-visualised (GitHub, accessed 2026-10-21). Numerical probes of Theorem 1.1 over primes $p\le 3\times10^7$, reporting that means of components 1–2 roughly agree with $\mathrm{PD}(1)$ while components 3–8 deviate, a drift the repository says resembles an unconverged limit over too short a range.
  • Perplexity / Orply, Prime Predecessor Factors Converge to the Poisson–Dirichlet Law (Orply, 2026-10-08). Explains the manuscript’s ranked logarithmic factor partition, equal weighting of primes, and size-biased route to the claimed limiting law. [AI-generated per source]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026,
  author = {{OpenAI}},
  title = {{The Poisson--Dirichlet law for prime predecessors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf}{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026}},
  year = {2026}
}

Family 011: Prime-factor statistics of $p-1$

Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

Other papers in this family: