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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 6, Pages 61–70 (Mi izv510)  

This article is cited in 5 scientific papers (total in 6 papers)

The matrix Euler–Fermat theorem

V. I. Arnol'd

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard–Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem.

DOI: https://doi.org/10.4213/im510

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English version:
Izvestiya: Mathematics, 2004, 68:6, 1119–1128

Bibliographic databases:

UDC: 511
MSC: 11A99, 15A36
Received: 02.03.2004

Citation: V. I. Arnol'd, “The matrix Euler–Fermat theorem”, Izv. RAN. Ser. Mat., 68:6 (2004), 61–70; Izv. Math., 68:6 (2004), 1119–1128

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. 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
    2. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. 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
    4. Mazur M., Petrenko B.V., “Generalizations of Arnold's version of Euler's theorem for matrices”, Japanese Journal of Mathematics, 5:2 (2010), 183–189  crossref  mathscinet  zmath  isi  scopus
    5. Marcin Mazur, B.V.. Petrenko, “Representations of analytic functions as infinite products and their application to numerical computations”, Ramanujan J, 2014  crossref  mathscinet  scopus
    6. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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