{"id":"openai-math/The-joint-Dickman-law-for-consecutive-integers","pdf":"https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026","tags":["analytic-number-theory","prime-factors","probability","dickman"],"score":0,"title":"The joint Dickman law for consecutive integers","domain":"Number theory","family":"012","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Independent largest prime factors of consecutive integers"}