Independent largest prime factors of consecutive integers at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Independent largest prime factors of consecutive integers

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	primes, fractions, patience	Levels:	1
Category:	Number theory	Lean version:	YES! ✔
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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.

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LEVEL 1 The joint Dickman law for consecutive integers

released 2026-09-24 | 3 theorems · 25 lemmas · 36 proofs · 33,931 words | PLAY LEVEL 1 » (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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