OpenAI · 011-01 · 2026-09-24 · Number theory

Weighted dilation graphs, smooth shifted primes and totient fibers

preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-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 $ε > 0$ .

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 $2 x < p \leq 5 x$ whose predecessors have no prime factor exceeding xδ.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

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.