Prime-factor distribution of prime predecessors
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For equally weighted primes p≤x, do the ordered prime factors of p−1, counted with multiplicity and normalized by their logarithms, approach Poisson–Dirichlet(1) in every fixed finite joint distribution?
Why it matters and what counts as progress
Why it matters: This tests whether the known factorization law for ordinary integers persists for shifted primes.
What counts as progress: An unconditional full joint-distribution theorem, rather than restricted mass or a conditional law; a largest-factor Dickman limit is a related narrower consequence.
How the problem developed
Kevin Ford, Sergei Konyagin and Florian Luca Recorded a conjecture that the normalized ordered prime factors of p−1, for prime p, follow the Poisson–Dirichlet law with parameter one.
- 24 Sept 2026 · Proof [2]
OpenAI Proved convergence of the normalized ordered prime factors of p−1 to PD(1) in every fixed finite joint distribution, resolving Ford–Konyagin–Luca's conjecture.
AI contribution
The Poisson-Dirichlet law for prime predecessors
For primes p≤x with equal weight, the normalized logarithms of the ordered prime factors of p−1, counted with multiplicity, converge in every fixed finite joint distribution to Poisson–Dirichlet(1). The manuscript also derives a fixed-threshold Dickman law for the largest factor.
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Publication dates and editorial changes.
OpenAI reports a Poisson–Dirichlet law for prime predecessors