Problem 928

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claims/: The 2 claim pages of Problem 928, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\alpha,\beta\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)<n^{\alpha}$ and $P(n+1)<(n+1)^\beta$ exist?

Formulation. The set with $P(n)\le n^{\alpha}$ and $P(n+1)\le n^{\beta}$ in place of the strict inequalities and the threshold $(n+1)^{\beta}$ differs from the problem's set, up to $X$, by at most $\pi(X^{\alpha})+\pi((X+1)^{\beta})$ integers, so a density statement for either set holds for the other; the claim page below states the bridge, which is elementary and not in Lean.

Status. Proved here; the site labels the problem OPEN (page last edited 3 April 2026). The OpenAI release of September 2026 claims that the density exists and equals $\rho(1/\alpha)\rho(1/\beta)$, with $\rho$ Dickman's function, the independence Erdős asked for, and proves the theorem in Lean. This corpus built that Lean with only the three standard axioms, its fingerprint identical to the release's comparator challenge, and the statement audit found it exact, with the elementary bridge to the Statement's strict inequalities stated on the claim page, so the claim is accepted on OpenAI 2026 and the problem stands solved; no outside review is known. Wang [Wa21] proved the density under the Elliott–Halberstam conjecture for friable integers, an accepted conditional claim, Wang 2021, which settles no standing. Teräväinen [Te18] proved the product law in logarithmic density, and Tao and Teräväinen (Algebra Number Theory 13 (2019), Remark 3.3) proved it for ordinary averages outside an exceptional set of scales of logarithmic density zero; neither settles an instance of the natural-density question, so neither has a claim page. The site's commentary also records Erdős's further question whether infinitely many such $n$ exist, which Meza observed follows from Schinzel's theorem [Sc67b] that $P(n(n+1))\le n^{O(1/\log\log n)}$ for infinitely many $n$; that answers a side question and settles no instance of the density question, so it has no claim page.

Source. erdosproblems.com/928, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #928, https://www.erdosproblems.com/928.

References.

• [Di30] K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astr. Fys. (1930), 1-14.
• [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
• [Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177-236.
• [Te18] Teräväinen, Joni, On binary correlations of multiplicative functions. Forum Math. Sigma (2018), Paper No. e10, 41.
• [Wa21] Wang, Zhiwei, Three conjectures on $P^+(n)$ and $P^+(n+1)$ hold under the Elliott-Halberstam conjecture for friable integers. J. Number Theory (2021), 1-11.

Formalization. No statement in formal-conjectures is recorded. The release proves its theorem in Lean (OAI.JointDickmanPaper.joint_law), pinned by its comparator challenge and linked from the claim page; this corpus built it from the pinned revision with the axioms propext, Classical.choice and Quot.sound only. The bridge from the theorem's thresholds to the Statement's is stated on the claim page, not in Lean.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• openai_2026_joint_dickman_law_consecutive_integers
• openai_2026_joint_dickman_law_consecutive_integers / theorem_1_1
• schinzel_nd_two_theorems_gelfond_applications
• teravainen_2018_binary_correlations_multiplicative_functions
• teravainen_2018_binary_correlations_multiplicative_functions / theorem_1_11
• teravainen_2018_binary_correlations_multiplicative_functions / theorem_1_14
• teravainen_2018_binary_correlations_multiplicative_functions / theorem_1_19