Prime-factor statistics of p-1

Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law \mathrm{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-\varepsilon} totient preimages, for every ε > 0.

Manuscripts (3)

Weighted dilation graphs, smooth shifted primes and totient fibers

September 24, 2026 · source · cite as \cite{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}

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-\varepsilon } preimages. We also show that, for every fixed δ > 0, there are at least x^{1-o(1)} primes p in 2x\lt p\le5x whose predecessors have no prime factor exceeding x^δ.

The Poisson-Dirichlet law for prime predecessors

September 24, 2026 · source · cite as \cite{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026}

For a prime p chosen uniformly from 3\le p\le x, list the prime factors of p-1 in decreasing order, with multiplicity. As x\to\infty, 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.

Prime Predecessors with an Even Number of Prime Factors

September 17, 2026 · source · cite as \cite{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}

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 \mu(p-1)=1.

Result 011 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.