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Mat. Zametki, 2010, Volume 88, Issue 4, Pages 594–604 (Mi mz8854)  

This article is cited in 7 scientific papers (total in 7 papers)

The Mean Number of Steps in the Euclidean Algorithm with Odd Incomplete Quotients

A. V. Ustinov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: The length of the continued-fraction expansion of a rational number with odd incomplete quotients is expressed via the Gauss–Kuzmin statistics for the classical continued fraction. This has made it possible to prove asymptotic formulas, similar to those already known for the classical Euclidean algorithm, for the mean length of the Euclidean algorithm with odd incomplete quotients.

Keywords: Euclidean algorithm, Gauss–Kuzmin statistics, continued-fraction expansion, dual fraction, incomplete quotient

DOI: https://doi.org/10.4213/mzm8854

Full text: PDF file (494 kB)
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English version:
Mathematical Notes, 2010, 88:4, 574–584

Bibliographic databases:

UDC: 517.524+510.52+519.712.61
Received: 13.04.2010

Citation: A. V. Ustinov, “The Mean Number of Steps in the Euclidean Algorithm with Odd Incomplete Quotients”, Mat. Zametki, 88:4 (2010), 594–604; Math. Notes, 88:4 (2010), 574–584

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. O. A. Gorkusha, “O srednei dline diagonalnykh drobei Minkovskogo”, Dalnevost. matem. zhurn., 11:1 (2011), 10–27  mathnet  elib
    2. A. V. Ustinov, “O statistikakh Gaussa — Kuzmina v korotkikh intervalakh”, Dalnevost. matem. zhurn., 11:1 (2011), 93–98  mathnet
    3. D. Frolenkov, “Asymptotic behaviour of the first moment of the number of steps in the by-excess and by-deficiency Euclidean algorithms”, Sb. Math., 203:2 (2012), 288–305  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Zhabitskaya E.N., “Continued fractions with odd partial quotients”, Int. J. Number Theory, 8:6 (2012), 1541–1556  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. V. Ustinov, “Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals”, Sb. Math., 204:5 (2013), 762–779  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. A. V. Ustinov, “Three-dimensional continued fractions and Kloosterman sums”, Russian Math. Surveys, 70:3 (2015), 483–556  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. V. A. Bykovskii, D. A. Frolenkov, “The average length of finite continued fractions with fixed denominator”, Sb. Math., 208:5 (2017), 644–683  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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