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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 1, Pages 1–15 (Mi faa92)  

This article is cited in 12 scientific papers (total in 13 papers)

Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes

V. I. Arnol'dab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Université Paris-Dauphine

Abstract: Congruences generalizing Fermat's little theorem are proved for the traces of powers of integer matrices. Their relations to Lobachevsky geometries over finite fields and combinatorics of the matrix squaring operation as well as to the corresponding Riemann surfaces with their Kepler cubes are discussed.

Keywords: arithmetics, symmetric function, de Sitter world, trace, Fermat's little theorem, Lobachevsky geometry, Kepler cube, Riemann surface

DOI: https://doi.org/10.4213/faa92

Full text: PDF file (215 kB)
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English version:
Functional Analysis and Its Applications, 2004, 38:1, 1–13

Bibliographic databases:

UDC: 51
Received: 03.10.2003

Citation: V. I. Arnol'd, “Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes”, Funktsional. Anal. i Prilozhen., 38:1 (2004), 1–15; Funct. Anal. Appl., 38:1 (2004), 1–13

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Arnol'd, “The matrix Euler–Fermat theorem”, Izv. Math., 68:6 (2004), 1119–1128  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. R. Alimov, “Connectedness of suns in the space $c_0$”, Izv. Math., 69:4 (2005), 651–666  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Arnold V., “Number-theoretical turbulence in Fermat-Euler arithmetics and large young diagrams geometry statistics”, J. Math. Fluid Mech., 7, suppl. 1 (2005), S4–S50  crossref  mathscinet  zmath  isi  scopus
    4. A. V. Zarelua, “On matrix analogs of Fermat's little theorem”, Math. Notes, 79:5 (2006), 783–796  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Arnold V.I., “On the matricial version of Fermat-Euler congruences”, Jpn. J. Math., 1:1 (2006), 1–24  crossref  mathscinet  zmath  isi  elib  scopus
    6. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. Huhtanen M., “Averaging operators for exponential splittings”, Numer. Math., 106:3 (2007), 511–528  crossref  mathscinet  zmath  isi  scopus
    8. Eagle L., “Commercial media literacy - What does it do, to whom-and does it matter?”, Journal of Advertising, 36:2 (2007), 101–110  crossref  mathscinet  isi  scopus
    9. A. V. Zarelua, “On Congruences for the Traces of Powers of Some Matrices”, Proc. Steklov Inst. Math., 263 (2008), 78–98  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    10. Mazur M., Petrenko B.V., “Generalizations of Arnold's version of Euler's theorem for matrices”, Jpn J Math, 5:2 (2010), 183–189  crossref  mathscinet  zmath  isi  scopus
    11. A. N. Abyzov, I. I. Mukhametgaliev, “On Some Matrix Analogs of the Little Fermat Theorem”, Math. Notes, 101:2 (2017), 187–192  mathnet  crossref  crossref  mathscinet  isi  elib
    12. Steinlein H., “Fermat'S Little Theorem and Gauss Congruence: Matrix Versions and Cyclic Permutations”, Am. Math. Mon., 124:6 (2017), 548–553  crossref  mathscinet  zmath  isi  scopus
    13. Maruyama F., Deguchi Y., Toyoizumi M., “On the Cardinality of Subsets of the Matrix Ring Over Certain Residue Ring”, JP J. Algebr. Number Theory Appl., 40:6 (2018), 1079–1087  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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