OpenAI · 011-02 · 2026-09-24 · Number theory

The Poisson-Dirichlet law for prime predecessors

preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026

Family 011. The Ford–Konyagin–Luca conjecture on prime predecessors.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves that the normalized ordered logarithms of the prime factors of $p - 1$ , counted with multiplicity, converge jointly to the Poisson–Dirichlet law $PD (1)$ 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 $n^{1 - ε}$ totient preimages, for every $ε > 0$ .

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

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.