Independence of largest prime factors in consecutive integers

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Independence of largest prime factors in consecutive integers

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Take any statement about the largest prime factors of n and n+1, and work out what density it should have, why the ordering P⁺(n) < P⁺(n+1) holds exactly half the time, and why decades of partial results never quite showed it.

The question and the law

For one number, Dickman's function answers everything: the proportion of n ≤ X with P⁺(n) ≤ X^a tends to ρ(1/a). For example, ρ(2) = 1 − log 2 ≈ 0.31 [0:14]. Neighbours n and n+1 are tightly linked (coprime, opposite parity), so Erdős and Pomerance asked whether U = log P⁺(n)/log n and V = log P⁺(n+1)/log n are independent in the limit [0:38].

A recent preprint claims Theorem 1.1, the joint Dickman law: for fixed a, b in (0,1), the natural density of n with P⁺(n) ≤ n^a and P⁺(n+1) ≤ n^b is ρ(1/a)·ρ(1/b). This is an ordinary limit over every large X [2:11]. Worked case: a = b = ½ gives ρ(2)² ≈ 0.094, about one n in eleven [2:41].

Why ½, and why it was new

In the unit square of (U, V), the product law gives the diagonal zero mass and is symmetric, so each ordering has density ½ (Corollary 1.2) [3:14]. Earlier work stopped short in three ways:

lower densities, from Erdős–Pomerance's ≈0.0099 up to Yang's 0.280. A floor never proves that a limit exists.
logarithmic density (Teräväinen). A log density does not imply a natural density.
ordinary averages with exceptional scales left out (Tao–Teräväinen) [1:13].

How the proof runs

The preprint swaps the largest prime for bin characters, which count prime factors in log-scale bins with unit phases. Fourier inversion reduces the claim to one mixed decorrelation, (1/x)Σ ḡ(n)F(n+1) → 0 [4:12]. Suppose it fails. A divisor-weight amplifier and Cauchy–Schwarz then remove g. A substitution turns the leftover energy into a shift graph, and cut-norm sampling turns that graph into a kernel of simple prime weights. The Matomäki–Radziwiłł short-interval theorem then forces cancellation, which is a contradiction [4:57].

When you read a claim of this kind, ask first: which density, and at which scales? Then multiply the Dickman marginals.

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