Wang 2026 exact formula erdos problem 1005

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theorem_1_2: The claimed main theorem of the 2026 Wang-Xie-Zhao preprint: for all sufficiently large n the Problem 1005 function equals van Doorn's upper bound U(n), which is m + 1, m + 2, m + 2, m + 4 for n = 4m + 0, 1, 2, 3; an AI-assisted preprint and a proof claim on the site's tab, compiled at statement depth only.

theorem_1_3: The claimed computer-assisted theorem of the 2026 Wang-Xie-Zhao preprint: for every n >= 4 the Problem 1005 function equals van Doorn's upper bound U(n), except that it is U(n) - 2 for n = 15, 27 and U(n) - 1 for thirteen further n up to 91; an AI-assisted preprint, compiled at statement depth only.

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Yanmohan Wang, Mingxu Xie and Ziyuan Zhao, An exact formula for Erdős' problem 1005, arXiv:2608.15681v1 [math.NT] (16 August 2026), 9 pages. The abstract page lists one version and no journal reference; no journal record was found (Crossref bibliographic query, 2026-09-18); Semantic Scholar lists no citing record. A preprint, not cited by the site's Problem 1005 page, which carries the first author's full-proof claim on its proof-claims tab (submitted 28 July 2026, before the arXiv posting, with the note that the proposed proof "has not yet been peer reviewed" and, in the tab's summary, that the argument "does not determine $n_0$").

The copy read for this card is the arXiv PDF of version 1 (9 pages, complete text layer; 350,292 bytes), read on the rendered page image of p. 1 and in the text layer elsewhere; its arXiv stamp, arXiv:2608.15681v1 [math.NT] 16 Aug 2026, is printed in the margin of p. 1 only (record https://arxiv.org/abs/2608.15681). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2608.15681), every other right reserved.

Provenance declared by the manuscript (p. 9, "AI-use declaration"): the authors used an AI system to help them generate candidate proof strategies, and write that "All AI-generated suggestions were verified and refined by the authors, who take full responsibility for the final content of the paper." The system is named there and not here. Code availability (p. 9): a public repository (dct-cell/erdos, folder 1005; its head 807b3d1d085532a1e4443cd1aa7bca7faa78f99c of 13 August 2026 was listed through the GitHub API on 2026-09-18: a README, the paper's PDF and TeX source and a verification folder; nothing was run or read beyond the listing).

Read status: claims checked for the abstract, the definition (1), Theorems 1.1--1.3 (read clause by clause on the page image of p. 1); the proofs (Sections 2--5) were read for their structure only, and no step was checked; nothing here is independently reviewed. This is a claim source: its statements are leads with provenance, not status.

Contents

• Section 1 (p. 1): $\mathcal F_n=(a_0/b_0,a_1/b_1,\ldots)$ the Farey sequence of order $n$; for $n\ge4$, $f(n):=\min\{j-i-1:0\le i<j<|\mathcal F_n|,\ (a_j-a_i)(b_j-b_i)<0\}$ (1), the intervening-fractions convention of the Cipollini preprint. History: Mayer, Erdős's $f(n)\ge n/400-O(1)$, van Doorn's $(1/12-o(1))n$ and van Doorn's four upper bounds, written as $U(n)=m+1,\ m+2,\ m+2,\ m+4$ for $n=4m+r$, $r=0,1,2,3$ (Theorem 1.1, attributed to van Doorn's Theorem 1); van Doorn's computation for $n\le5000$ and conjecture $f(n)=U(n)$ for $n\ge92$; Cipollini's $f(n)=(1/4+o(1))n$. Theorem 1.2: $f(n)=U(n)$ for all sufficiently large $n$. Theorem 1.3 (computer-assisted): with $\mathcal E=\{7,9,11,15,19,23,25,27,31,35,39,49,51,63,91\}$, for every $n\ge4$, $f(n)=U(n)-2$ for $n\in\{15,27\}$, $U(n)-1$ for $n\in\mathcal E\setminus\{15,27\}$, and $U(n)$ otherwise.
• Section 2 (pp. 2--4): auxiliary functions and tools: $\Phi(m)=\sum_{r\le m}\varphi(r)$ with $\Phi(m)\ge\frac27m(m+1)$ for every $m\ge1$ (Lemma 2.1, its cases $m<360$ by the computation of Section 5); the weighted totient function $G(x)=\sum_{1\le r<x}\frac{\varphi(r)}r(1-\frac rx)$ of the Cipollini preprint, with $G(x+1)-G(x)\ge5/18$ for $x\ge1$ among its properties (Lemma 2.2); a counting function $\mathcal H_t(X,Y)$ written through $G$ (Lemma 2.3); Dirichlet's approximation theorem (Theorem 2.4) and Dress's discrepancy bound for $|\mathcal F_n\cap(0,\alpha]|$ (Theorem 2.5).
• Section 3 (pp. 4--6): Theorem 3.1, for $n\ge4$ and $a/b\in\mathcal F_n$ with $1\le a\le b-2$ and $b-2a\notin\{1,2\}$, at least $5n/18-O(\sqrt n\log n)$ fractions of $\mathcal F_n$ in $(a/b,(a+1)/(b-1))$; Lemmas 3.2 and 3.3 treat $b>n/4$.
• Section 4 (pp. 7--8): Theorem 4.1, for $m\ge3$, $r\in\{0,1,2,3\}$, $n=4m+r$ and $a/b\in\mathcal F_n$ with $1\le a\le b-2$ and $b-2a\in\{1,2\}$ (and $a/b\ne(2m+1)/(4m+3)$ when $r=3$), at least $U(n)$ fractions of $\mathcal F_n$ in $(a/b,(a+1)/(b-1))$, by explicit lists; then the proof of Theorem 1.2 (p. 8).
• Section 5 (pp. 8--9): the proof sketch of Theorem 1.3, with the ranges $5000\le n\le83387$, $83388\le n\le181854$ and $181855\le n\le5504797$ verified by program and $n\ge5504798$ covered by the analytic estimate, together with a direct enumeration of $\mathcal F_n$ for $4\le n\le5000$; then the code availability statement and the AI-use declaration (p. 9).

Compiled scope

The statements were read; Theorems 1.2 and 1.3 are compiled as claim pages. The proofs were read for their structure only and no step was checked; the program was not run, and nothing is independently reviewed.

Bears on. #1005, as a lead beyond the asymptotic answer: the claim that van Doorn's upper bound $f(n)\le\lfloor n/4\rfloor+d$ is attained for all sufficiently large $n$ (Theorem 1.2) and, with a finite computation, a claimed exact value of $f(n)$ for every $n\ge4$, the bound being attained except at fifteen listed $n\le91$ (Theorem 1.3); an unrefereed AI-assisted preprint and a proof claim on the site's tab, not a status source.

Results.

• Theorem 1.2 (p. 1): $f(n)=U(n)$ for all sufficiently large $n$, where $U(4m)=m+1$, $U(4m+1)=U(4m+2)=m+2$, $U(4m+3)=m+4$.
• Theorem 1.3 (p. 1): the exact value of $f(n)$ for every $n\ge4$: $U(n)-2$ for $n\in\{15,27\}$, $U(n)-1$ on the rest of $\mathcal E=\{7,9,11,15,19,23,25,27,31,35,39,49,51,63,91\}$, and $U(n)$ otherwise (computer-assisted).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.