OpenAI · 012-01 · 2026-09-24 · Number theory
The joint Dickman law for consecutive integers
preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026
Family 012. Independent largest prime factors of consecutive integers.
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
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of and are asymptotically independent in ordinary natural density. In particular, the integers satisfying have density .
Let denote the largest prime factor of n. We prove that and are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering has natural density 1/2.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 012: Independent largest prime factors of consecutive integers. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.