Skip to main content
SpringerLink
Log in

Fermat–Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions

  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

Let n be an integer. A Fermat–Euler dynamical system acts on the set of mod-n residues coprime to n by multiplication by a constant (which is also coprime to n). We study the dependence of the period and the number of orbits of this dynamical system on n. Theorems generalizing Fermat's little theorem, as well as empirical conjectures, are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Arnold, Arithmetics of Binary Quadratic Forms, Symmetry of Their Continued Fractions, and Geometry of Their de Sitter World, Moscow, Dubna, MCNMO, 2002, pp. 1–40. (Bol. Soc. Brasil. Mat., 2003.)

    Google Scholar 

  2. V. I. Arnold, “Weak asymptotics for the number of decompositions of Diophantine problems,” Funkts. Anal. Prilozhen.,33, No. 4, 65–66 (1999).

    Google Scholar 

  3. I. V. Arnold, Number Theory [in Russian], Moscow, Uchpedgiz, 1938.

    Google Scholar 

  4. B. A. Venkov, Elementary Number Theory [in Russian], ONTI NKTP, 1937.

  5. V. I. Arnold, Euler Groups and Arithmetics of Geometric Progressions [in Russian], Moscow, MCCME, 2003.

    Google Scholar 

  6. V. Arnold, “Ergodic and arithmetic properties of geometrical progression's dynamics and of its orbits,” Moscow Mathematical Journal, 3, 1–20 (2003).

    Google Scholar 

  7. V. I. Arnold, “Topology and statistics of the formulae of arithmetics,” Usp. Mat. Nauk, 58 (2003).

  8. V. I. Arnold, “Topology of algebra: combinatorics of the squaring operation,” Funkts. Anal. Prilozhen., 37 (2003).

Download references

Author information

Authors and Affiliations

  1. V. A. Steklov Mathematical Institute, Russian Academy of Sciences; CEREMADE, Université Paris-Dauphine, France

    V. I. Arnold

Authors
  1. V. I. Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arnold, V.I. Fermat–Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions. Functional Analysis and Its Applications 37, 1–15 (2003). https://doi.org/10.1023/A:1022915825459

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022915825459

Search

Navigation