OpenAI research collection · 1 manuscript · Source checked 2026-10-19
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.
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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.
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