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Erdos Problem #371: do the integers with P(n) < P(n+1) have density 1/2? (Erdos-Pomerance joint independence of largest prime factors)

Erdős problem #371 · erdosproblems.com/371

Let P+(n)P^+(n) be the largest prime factor of nn and ρ\rho the Dickman function, so that P+(n)≤naP^+(n)\le n^a has density ρ(1/a)\rho(1/a). Erdos and Pomerance (1978) conjectured that the largest prime factors of nn and n+1n+1 behave independently, so that P+(n)≤naP^+(n)\le n^a and P+(n+1)≤nbP^+(n+1)\le n^b hold simultaneously with density ρ(1/a)ρ(1/b)\rho(1/a)\rho(1/b), and in particular that the set of nn with P+(n)<P+(n+1)P^+(n)<P^+(n+1) has natural density 1/21/2 (Erdos Problem #371). They proved positive lower density for each ordering; later bounds reached about 0.280.28. Teravainen (2018) proved the logarithmic-density version and Tao and Teravainen (2019) the natural-density version at almost all scales. Does {n:P+(n)<P+(n+1)}\{n:P^+(n)<P^+(n+1)\} have natural density 1/21/2, and are log⁡P+(n)/log⁡n\log P^+(n)/\log n and log⁡P+(n+1)/log⁡n\log P^+(n+1)/\log n asymptotically independent?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Analytic number theory; smooth numbers and largest prime factors
Posed by
P. Erdos and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. (1978), p. 311; Erdos Problem #371
Year posed
1978
Years open
48y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for fixed a,b∈(0,1)a,b\in(0,1), 1X#{2≤n≤X:P+(n)≤na,P+(n+1)≤nb}→ρ(1/a)ρ(1/b)\frac1X\#\{2\le n\le X:P^+(n)\le n^a,P^+(n+1)\le n^b\}\to\rho(1/a)\rho(1/b), unconditionally and at every large scale. Corollary 1.2: P+(n)<P+(n+1)P^+(n)<P^+(n+1) and the reverse ordering each have natural density 1/21/2. Inclusion-exclusion also gives the Erdos-Pomerance upper-tail independence statement. It gives no rate and no uniformity in a,ba,b (unlike the almost-all-scales quantitative law of Tao-Teravainen 2026), and says nothing about three consecutive integers (Erdos #372 and Balog's density 1/6 conjecture).

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against Erdos Problem #371 and the Erdos-Pomerance independence conjecture. The Lean challenge lean/ComparatorChallenges/JointDickman.json (solution module OAI/NumberTheory/JointDickman/PaperMain.lean present at the pinned commit) is not in the formalization catalogue; its statement was read here. joint_law states that for 0<a,b<10<a,b<1 the natural density of nn with P+(n)≤naP^+(n)\le n^a and P+(n+1)≤nbP^+(n+1)\le n^b tends to ρ(1/a)ρ(1/b)\rho(1/a)\rho(1/b), with ρ\rho built from the delay equation, and increasing_order and decreasing_order state density 1/21/2 for each strict ordering. That is the headline. Not rebuilt here. The result is qualitative: no error term, fixed a,ba,b only.

Sources

Changelog1 change

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