MathAudit

Paper metadata

Original submission
February 8, 2020
Latest revision
January 31, 2021 · version 2
Claimed MSC
11A55
arXiv categories
math.HO, math.NT
Comments
7 pages, in Russian, submitted to "Istoriko-matematicheskie issledovaniya", minor corrections

Abstract

This is a brief historical note about famous Legendre's criterium for convergent of continued fraction expansion. The paper is written in Russian.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements3 reported findingsCorrect

The historical account and the stated Legendre convergent criteria are mathematically correct; a single summation index in the Farey count is typographical.

Legendre criterionCorrect

The necessary-and-sufficient convergent criterion and its classical corollary are correct

Pages 2–6 · criteria and proofs · arXiv:2002.07587v2

The neighboring Farey fractions characterize the interval of real numbers for which a reduced rational is a convergent. Comparing the endpoint distances gives the familiar sufficient bound ∣α−p/q∣<1/(2q2)|\alpha-p/q|<1/(2q^2) and the sharper exact criterion discussed in the sources.

Full paper, version 2 ↗
Farey enumerationTypo · no status impact

The totient summation uses the wrong dummy variable

Page 6 · Farey-sequence count · arXiv:2002.07587v2

The number of reduced fractions of order at most qq is summed as ∑k≤qφ(k)\sum_{k\leq q}\varphi(k); the printed summand φ(q)\varphi(q) repeats the upper endpoint. The surrounding count and asymptotic use the standard corrected sum.

Legendre's sufficient boundCorrect

The classical half-square criterion follows from the exact interval

Pages 4–6 · Farey-neighbor interval and Legendre corollary · arXiv:2002.07587v2

For a reduced fraction p/qp/q, its two neighboring fractions of order qq have determinant one with p/qp/q. Their endpoint distances therefore bound the interval on which p/qp/q is a continued-fraction convergent. The inequality ∣α−p/q∣<1/(2q2)|\alpha-p/q|<1/(2q^2) lies inside that interval, which proves the stated sufficient criterion.

02Proofs1 reported findingCorrect

The elementary continued-fraction and Farey-neighbor arguments are correct and complete after the dummy-index correction.

Sections 2–5Correct and complete

The interval characterization proves each quoted criterion

Pages 2–7 · historical formulations and derivations · arXiv:2002.07587v2

The determinant-one relation for neighboring fractions gives the exact endpoint distances, and the continued-fraction recursion identifies the relevant neighbors. All displayed implications then follow directly.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2002.07587v2
Authors listed
N. G. Moshchevitin, A. Yu. Yashnikova
Audit date
August 20, 2026
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