The Ford-Konyagin-Luca conjecture: the prime factors of follow the Poisson-Dirichlet law
For a prime , list the prime factors of with multiplicity in decreasing order and set . For a uniformly random integer the analogous normalized factor sizes converge to the Poisson-Dirichlet law (Billingsley, Donnelly-Grimmett). Ford, Konyagin and Luca (2010, Section 6, Conjecture 5) conjectured the same for shifted primes. The first coordinate already contains the shifted-prime Dickman law stated by Granville, for fixed ; unconditionally only lower bounds of size for were known (Baker-Harman, Lichtman), and the full law was known under Elliott-Halberstam (Bharadwaj-Rodgers). With uniform among primes up to , does converge in finite-dimensional distributions to as ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Analytic number theory; smooth shifted primes
- Posed by
- Kevin Ford, Sergei Konyagin and Florian Luca (Prime chains and Pratt trees, GAFA 2010, Section 6, Conjecture 5)
- Year posed
- 2010
- Years open
- 16y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every and bounded continuous on , the average of over primes tends to with , unconditionally and with equal weight on each prime. The manuscript notes this gives Granville's fixed- asymptotic , so a positive proportion of primes have -smooth predecessors for each fixed . No error terms or uniformity as grows are claimed. The companion on (infinitely many primes with squarefree with an even number of prime factors) is linked here, not cataloged, since it cites no posing source.
What the AI did
The release README says all results in the release were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript builds on graph and ideal-kernel estimates proved in the companion 'Weighted dilation graphs, smooth shifted primes and totient fibers' (same date, separate entry for Erdos Problem #821); a third companion (17 September) uses the same machinery for primes with mu(p-1) = 1.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and the history section were read against Conjecture 5 of Ford-Konyagin-Luca as the manuscript quotes it (same multiplicity convention, normalization and equal weighting of primes). No Lean formalization exists for this family. The claim is unconditional and reaches what was previously known only under the Elliott-Halberstam conjecture; the manuscript says that unconditional distribution results (level one half) cover only part of the simplex and that its extraction argument reaches every compact subset with coordinate sum below one. A claim of this strength should be read as unverified until specialists examine it. It imports estimates from the companion manuscript. Not refereed here.