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Independent largest prime factors of consecutive integers

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)\lt P^+(n+1)

have density 1/2.

Manuscripts in this family · 1

The joint Dickman law for consecutive integers

Sep 24, 2026·paper.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)\lt P^+(n+1)

has natural density 1/2.

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