Prime-factor statistics of p − 1
Prime-factor statistics of p − 1 is a family of three manuscripts dated September 2026 in OpenAI's public mathematics collection (github.com/openai/math). They concern the prime factors of the predecessor p − 1 of a prime p, and one of them addresses Erdős's conjecture on the largest fibres of Euler's totient function (Erdős Problem #821). This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original results
- Totient fibres and smooth shifted primes. For every ε > 0, infinitely many positive integers n have more than n1−ε solutions m of φ(m) = n. The arithmetic input is that, for every fixed 0 < δ < 1/4, at least x1−o(1) primes p with 2x < p ≤ 5x have every prime factor of p − 1 at most xδ; the manuscript notes that the count also holds for every fixed δ > 0. Source: Weighted dilation graphs, smooth shifted primes and totient fibers (24 September 2026).
- Poisson–Dirichlet law. For a prime p chosen uniformly from 3 ≤ p ≤ x, the logarithms of the prime factors of p − 1 (in decreasing order, with multiplicity), divided by log(p − 1), converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. The manuscript states that this proves a conjecture of Ford, Konyagin and Luca. Source: The Poisson-Dirichlet law for prime predecessors (24 September 2026).
- Even number of prime factors. There are infinitely many primes p for which p − 1 is squarefree with an even number of prime factors, that is, μ(p − 1) = 1. Source: Prime predecessors with an even number of prime factors (17 September 2026).
- Formalization. The collection's index (commit
adc7f12) lists no Lean formalization for this family. - Earlier results, as summarized in the totient-fibre manuscript. Pomerance (1980) formulated the conjecture as C = 1, where C is the supremum of exponents c with g(n) > nc infinitely often, and attributed it to Erdős. Baker and Harman (1998) obtained smoothness exponent 0.2961 for shifted primes and totient multiplicity exponent 0.7039. Lichtman (2022) proved a lower bound for every smoothness exponent above 15/(32√e) = 0.2843… and obtained multiplicity exponent 0.7156. Bharadwaj and Rodgers (2026) prove the full Poisson–Dirichlet law for shifted primes under the Elliott–Halberstam conjecture.
- Problem database. The erdosproblems.com entry for Problem 821 (page last edited 1 October 2025) records Lichtman's exponent 0.71568… as the best known bound (checked 14 October 2026).
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | J. Guo, aster2024/openai-math-followups (GitHub), consequences list, item 2 | Use (conditional consequence) | Conditional on Theorem 1.1 of the Poisson–Dirichlet manuscript, and through results of Alford, Granville and Pomerance and of Kalmynin, lower bounds for the counts of Carmichael numbers (x5/12−ε, and x0.4736−ε with Harman's admissible value) and for Novak–Carmichael numbers (x1−o(1)). | The repository states that the list was compiled with AI systems under the author's direction and that the author takes responsibility for its content. |
| 2026-10-14 | Yao Xu, cadamcat/totient-fibers-lean4 (GitHub; commit dfce357) |
Formalization (Lean) | A Lean 4 formalization of Erdős Problem 821 that follows the totient-fibre manuscript, with its target statement aligned to the Formal Conjectures statement of the problem. The README states that the proofs are verified by the Lean 4 kernel and describes a verification script that compares the statement, scans for forbidden constructs and checks the axioms used. | Author-reported; the README states that the work was developed with AI assistance. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.