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Mat. Sb., 2009, Volume 200, Number 4, Pages 131–160 (Mi msb6547)  

This article is cited in 16 scientific papers (total in 16 papers)

The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments

A. V. Ustinov

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

Abstract: It is shown that on the average the Frobenius numbers $f(a,b,c)$ behave like $\frac8\pi\sqrt{abc}$ .
Bibliography: 28 titles.

Keywords: Frobenius numbers, continued fractions, Kloosterman sums.

DOI: https://doi.org/10.4213/sm6547

Full text: PDF file (629 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:4, 597–627

Bibliographic databases:

UDC: 511.336+511.218+511.335
MSC: 11D85, 11D04
Received: 05.08.2008 and 05.01.2009

Citation: A. V. Ustinov, “The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments”, Mat. Sb., 200:4 (2009), 131–160; Sb. Math., 200:4 (2009), 597–627

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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. Aliev I., Henk M., “Integer knapsacks: average behavior of the Frobenius numbers”, Math. Oper. Res., 34:3 (2009), 698–705  crossref  mathscinet  zmath  isi  elib  scopus
    2. Shchur V., Sinai Ya., Ustinov A., “Limiting distribution of Frobenius numbers for $n=3$”, J. Number Theory, 129:11 (2009), 2778–2789  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. V. Ustinov, “On the distribution of Frobenius numbers with three arguments”, Izv. Math., 74:5 (2010), 1023–1049  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. I. S. Vorobev, “Eksperimentalnoe issledovanie problemy Frobeniusa dlya trekh argumentov”, Dalnevost. matem. zhurn., 11:1 (2011), 3–9  mathnet  elib
    5. A. V. Ustinov, “O statistikakh Gaussa — Kuzmina v korotkikh intervalakh”, Dalnevost. matem. zhurn., 11:1 (2011), 93–98  mathnet
    6. A. A. Illarionov, “The average number of local minima of three-dimensional integer lattices”, St. Petersburg Math. J., 23:3 (2012), 551–570  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. A. V. Ustinov, “Geometric proof of Rødseth's formula for Frobenius numbers”, Proc. Steklov Inst. Math., 276 (2012), 275–282  mathnet  crossref  mathscinet  isi  elib  elib
    8. A. A. Illarionov, “The average number of relative minima of three-dimensional integer lattices of a given determinant”, Izv. Math., 76:3 (2012), 535–562  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. D. Frolenkov, “The mean value of Frobenius numbers with three arguments”, Izv. Math., 76:4 (2012), 760–819  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Shparlinski I.E., “Modular hyperbolas”, Jap. J. Math., 7:2 (2012), 235–294  crossref  mathscinet  zmath  isi  elib  scopus
    11. Strömbergsson A., “On the limit distribution of Frobenius numbers”, Acta Arith., 152:1 (2012), 81–107  crossref  mathscinet  zmath  isi  elib  scopus
    12. V. I. Bernik, A. V. Ustinov, “O raspredelenii tochek modulyarnoi giperboly”, Dalnevost. matem. zhurn., 14:2 (2014), 141–155  mathnet
    13. W.M.. Schmidt, “Integer matrices, sublattices of
      $$\mathbb {Z}^{m}$$
      Z m , and Frobenius numbers”, Monatsh Math, 2015  crossref  mathscinet  elib  scopus
    14. 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
    15. I. S. Vorob'ev, “On the Frobenius problem for three arguments”, Sb. Math., 207:6 (2016), 816–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. V. M. Fomichev, “Computational complexity of the original and extended Diophantine Frobenius problem”, J. Appl. Industr. Math., 11:3 (2017), 334–346  mathnet  crossref  crossref  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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