On the matricial version of Fermat–Euler congruences

  • V. I. Arnold
Article

Abstract.

The congruences modulo the primary numbers n=pa are studied for the traces of the matrices An and An-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 

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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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