On the matricial version of Fermat–Euler congruences

  • V. I. Arnold


The congruences modulo the primary numbers n=p a are studied for the traces of the matrices A n and A n-φ(n), where A is an integer matrix and φ(n) is the number of residues modulo n, relatively prime to n.

We present an algorithm to decide whether these congruences hold for all the integer matrices A, when the prime number p is fixed. The algorithm is explicitly applied for many values of p, and the congruences are thus proved, for instance, for all the primes p ≤ 7 (being untrue for the non-primary modulus n=6).

We prove many auxiliary congruences and formulate many conjectures and problems, which can be used independently.

Keywords and phrases.

Young diagram Newton–Girard formula multinomial coefficients Cesaro averaging symmetric functions finite Lobachevsky plane Vieta mapping Euler zeta function Euler group little Fermat Theorem geometric progression arithmetical turbulence 

Mathematics Subject Classification (2000).

05A10 05A17 11A15 11B50 11T60 51E20 51E25 


  1. 1.
    F. Aicardi, Empirical estimates of the average orders of orbits period lengths in Euler groups, C. R. Acad. Sci. Paris Sér. I, 339 (2004), 15–20.MATHMathSciNetGoogle Scholar
  2. 2.
    V.I. Arnold, Matrix Fermat theorem, finite circles and finite Lobachevsky plane, Funct. Anal. Appl., 38 (2004), 1–15.MATHCrossRefGoogle Scholar
  3. 3.
    V.I. Arnold, Fermat dynamics of matrices, finite circles and finite Lobachevsky planes, Cahiers du Ceremade, Univ. Paris-Dauphine No. 0434, 3 juin 2004, 31 pp.Google Scholar
  4. 4.
    A. Girard, Sur des découvertes nouvelles en algèbre, Amsterdam, 1629.Google Scholar
  5. 5.
    I. Newton, Arithmetica Universalis, Cambridge, 1707, 57–63.Google Scholar
  6. 6.
    T. Schönemann, Theorie der symmetrischen Functionen der Wurzeln einer Gleichnung, J. Reine Angew. Math. (Crelle) 19 (1839), 289–308.MATHCrossRefGoogle Scholar
  7. 7.
    T. Schönemann, Theorie der symmetrischen Functionen der Wurzeln einer Gleichnung II, J. Reine Angew. Math. (Crelle), 1846, 288.Google Scholar
  8. 8.
    C.J. Smith, A coloring proof of a generalization of Fermat’s little theorem, Amer. Math. Monthly, 93 (1986), 469–471.MathSciNetCrossRefGoogle Scholar
  9. 9.
    T. Szele, Une généralisation de la congruence de Fermat. (French), Mat. Tidsskr. B., 1948, 57–59.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer-Verlag 2006

Authors and Affiliations

  • V. I. Arnold
    • 1
    • 2
  1. 1.CEREMADEUniversite Paris 9 - DauphineParis cedex 16France
  2. 2.Steklov Mathematical InstituteMoscowRussia

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