Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 012-01. The joint Dickman law for consecutive integers
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OpenAI@2026-10-07#rem:oai-p-012-01
Authors: OpenAI. Family 012.
Title: The joint Dickman law for consecutive integers.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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