Independent largest prime factors of consecutive integers

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P^+(n)\lt P^+(n+1) have density 1/2. (Lean)

Manuscript

The joint Dickman law for consecutive integers

September 24, 2026 · source · cite as \cite{OAI:The-joint-Dickman-law-for-consecutive-integers-September-24-2026}

Let P^+(n) denote the largest prime factor of n. We prove that \log P^+(n)/\log n and \log P^+(n+1)/\log n are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering P^+(n)\lt P^+(n+1) has natural density 1/2.

Result 012 · 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.