Independent largest prime factors of consecutive integers

What OpenAI claims

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) < P⁺(n+1) have density 1/2. (Lean)

Paper

1. The joint Dickman law for consecutive integers

   24 September 2026 · Preprint, not peer reviewed · PDF

   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) < P⁺(n+1) has natural density 1/2.

Status

Preprint. The paper has 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.