Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 371

claims/: The 2 claim pages of Problem 371, one per claimant's result; the problem's standing derives from them.

Statement. Let $P(n)$ denote the largest prime factor of $n$ . Show that the set of $n$ with $P(n)<P(n+1)$ has density $1/2$ .

Status. Proved here; the site's label is OPEN (page last edited 23 January 2026). The OpenAI release's manuscript The joint Dickman law for consecutive integers (2026-09-24) claims that the normalized largest prime factors of $n$ and $n+1$ have independent Dickman limit laws in natural density and deduces from that the density $1/2$ this problem asks for. Its Lean proof of that corollary was built by this corpus with only the three standard axioms, its fingerprint identical to the release's comparator challenge, and the statement audit found it exact, so the claim is accepted on the release's joint Dickman law and the problem stands solved; the manuscript has no outside review. Wang [Wa21] proved the density under the Elliott–Halberstam conjecture for friable integers, an accepted conditional claim, Wang 2021, which settles no standing.

Source. erdosproblems.com/371, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #371, https://www.erdosproblems.com/371.

References.

[Er79e] Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82.
[ErPo78] Erdős, Paul and Pomerance, Carl, On the largest prime factors of $n$ and $n+1$ . Aequationes Math. (1978), 311-321.
[LuWa25] [[../library/arithmetic_functions/lu_2025_largest_prime_factors_consecutive_integers/_index|Lü, Xiaodong and Wang, Zhiwei, On the largest prime factors of consecutive integers]]. Monatsh. Math. (2025), 403-418.
[TaTe19] Tao, Terence and Teräväinen, Joni, The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures. Algebra Number Theory (2019), 2103-2150.
[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. Statement in formal-conjectures. The release's declaration OAI.JointDickmanPaper.increasing_order, pinned by its comparator challenge JointDickman.lean, proves the Statement; this corpus built it from the pinned revision with the axioms propext, Classical.choice and Quot.sound only, as the claim page records.

Current assessment

The Statement asks for the natural density of the $n$ with $P(n)<P(n+1)$ , asserted to be $1/2$ . The standing is solved through one accepted full claim, the OpenAI release's corollary of its joint Dickman law, recorded on its claim page. Its acceptance rests on the formalization: this corpus built the release's Lean declaration from the pinned revision with the three standard axioms only, its comparator fingerprint was identical, and the statement audit found it exact for the Statement. No outside review or refereed publication exists, and the site's page, last edited 23 January 2026, does not mention the release. Wang's conditional result has an accepted claim page of scope conditional, which the derivation does not count. The search behind this page is the site record of 2026-09-04 and the release at its pinned revision; no wider literature search is recorded. The results under Progress are cited from the site's commentary, the carded papers and the arXiv records named, without a check of their proofs.

Progress

The literature before the release settled the density in weaker senses only. Erdős and Pomerance conjectured the statement and proved that each strict ordering of $P(n)$ and $P(n+1)$ holds on a set of positive lower density (Erdős and Pomerance 1978); the lower bound was raised by several authors to $0.2017$ for each ordering (Lü and Wang 2025), then by Yang to $0.280$ for the $n$ with $P(n)<P(n+1)$ (arXiv:2607.16032, 2026-07-17; the release's manuscript cites the bound as holding for both orderings) and to $0.299$ for the same ordering (arXiv:2608.13299, 2026-08-13), the best unconditional bounds before the release; a forum comment of 2026-07-20 reports both. These bounds settle no instance of the question and have no claim pages. Teräväinen proved that the logarithmic density is $1/2$ (Teräväinen 2018), and Tao and Teräväinen that the natural density is $1/2$ at all scales outside an exceptional set of logarithmic density zero (Tao and Teräväinen 2019); Wang [Wa21] obtained the natural density under the Elliott–Halberstam conjecture for friable integers (Wang 2021). The release's accepted result of 2026 removes the averaging, the exceptional scales and the hypothesis.

Known Results

The site's commentary records, besides the results above, Erdős's further question [Er79e] whether for every $\alpha$ the set of $n$ with $P(n+1)>P(n)n^{\alpha}$ has a density, and Teräväinen's logarithmic-density answer to it with the value given by an explicit double integral against the Dickman function. The nontrivial neighbors are Problem 372 (three consecutive largest prime factors in decreasing order) and Problem 928 (the joint distribution of $P(n)$ and $P(n+1)$ , which the same release manuscript addresses).

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.

Linked from (21)

Wang's density one half under Elliott–Halberstam for friable integers OpenAI's joint Dickman law gives density one half problems/arithmetic_functions/E0371/claims Wang's joint Dickman law under Elliott–Halberstam for friable integers Joint Dickman law for consecutive integers Arithmetic Functions Arithmetic Functions arithmetic_functions/erdos_1978_largest_prime_factors Corollary (p. 319): the n with P(n) > P(n+1) have lower density greater than 0.0099 Theorem 1 (pp. 311--312): P(n) and P(n+1) are rarely within a factor x^δ of each other arithmetic_functions/lu_2025_largest_prime_factors_consecutive_integers Theorem 1 (p. 4): more than 0.2017x of the n < x have P+(n) < P+(n+1), and the same for P+(n) > P+(n+1)arithmetic_functions/openai_2026_joint_dickman_law_consecutive_integers Corollary 1.2: P+(n) < P+(n+1) has natural density 1/2 Theorem 1.1: the joint Dickman law for P+(n) and P+(n+1) in natural density arithmetic_functions/tao_2019_structure_correlations_multiplicative_functions_at_almost Corollary 1.16 (p. 9): P^+(n) < P^+(n+1) for half of all n at almost all scales arithmetic_functions/teravainen_2018_binary_correlations_multiplicative_functions Theorem 1.16: the n with P(n) < P(n+1) have logarithmic density 1/2 Theorem 1.17: logarithmic density of n with P(n+1) > P(n) n^alpha divisors/erdos_1979_unconventional_problems_number_theory_asterisque

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