Prime-factor statistics of p-1

What OpenAI claims

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 → ∞. 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.

Papers (3)

1. Weighted dilation graphs, smooth shifted primes and totient fibers

   24 September 2026 · Preprint, not peer reviewed · PDF

   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 n^1-ε preimages. We also show that, for every fixed δ > 0, there are at least x^1-o(1) primes p in 2x < p ≤ 5x whose predecessors have no prime factor exceeding xδ.

2. The Poisson-Dirichlet law for prime predecessors

   24 September 2026 · Preprint, not peer reviewed · PDF

   For a prime p chosen uniformly from 3 ≤ p ≤ x, list the prime factors of p-1 in decreasing order, with multiplicity. As x → ∞, their logarithms, divided by 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.

3. Prime Predecessors with an Even Number of Prime Factors

   17 September 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.