Result 011, Number theory

Prime-factor statistics of p−1p-1

Proves that the normalized ordered logarithms of the prime factors of p−1p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law PD(1)\mathrm{PD}(1) as p ranges uniformly over primes up to x and x→∞x\to\infty. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n1−εn^{1-\varepsilon} totient preimages, for every ε > 0.

Proof

The bigger picture

Why it matters

Subtracting one from a prime produces a number whose prime factors can have very different sizes. The manuscript claims that their relative sizes follow a precise statistical law, revealing order across these factorizations.

What changes?

Choose a prime p uniformly among primes from 3 to x. List the prime factors of p-1 from largest to smallest, keeping repeated factors, and divide each factor's logarithm by the logarithm of p-1. These shares sum to one. The manuscript reports that, as x grows without bound, every fixed finite collection of leading shares converges jointly in distribution to the Poisson-Dirichlet law with parameter one, a probability law for ranked shares. This would resolve the Ford-Konyagin-Luca conjecture.

What does that help mathematicians do?

The claimed law describes how several large prime factors share the logarithmic size of p-1, rather than describing only its largest factor. Researchers could use this joint distribution to obtain limiting statistics for the largest few shares together. This matters because it captures their dependence: knowing one factor's share constrains the space left for others. It is an asymptotic statistical statement, not a factorization method for individual primes.

Are there practical applications?

The immediate value is foundational, including understanding how many integers can share a totient value. Euler's totient counts the positive integers up to a given integer that are coprime to it. A related manuscript claims that, for every positive epsilon, infinitely many positive integers n have more than n to the power (1 minus epsilon) totient preimages. This quantifies how extremely many-to-one the totient function can be.

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.

3 manuscripts

Weighted dilation graphs, smooth shifted primes and totient fibers

September 24, 2026 68 pages

We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than n1−εn^{1-\varepsilon } preimages. We also show that, for every fixed δ > 0, there are at least x1−o(1)x^{1-o(1)} primes p in 2x<p≤5x2x\lt p\le5x whose predecessors have no prime factor exceeding xδ.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026,
  author = {{OpenAI}},
  title = {{Weighted dilation graphs, smooth shifted primes and totient fibers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf}{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}},
  year = {2026}
}

The Poisson-Dirichlet law for prime predecessors

September 24, 2026 74 pages

For a prime p chosen uniformly from 3≤p≤x3\le p\le x, list the prime factors of p−1p-1 in decreasing order, with multiplicity. As x→∞x\to\infty, their logarithms, divided by log⁡(p−1)\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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Prime Predecessors with an Even Number of Prime Factors

September 17, 2026 45 pages

We prove that there are infinitely many primes p for which p−1p-1 is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with μ(p−1)=1\mu(p-1)=1.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026,
  author = {{OpenAI}},
  title = {{Prime Predecessors with an Even Number of Prime Factors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf}{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.