Skip to main content
SpringerLink
Log in

Une application des nombres de Pisot à l'algorithme de Jacobi-Perron

An application of the Pisot numbers to the Jacobi-Perron algorithm

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

SinceO. Perron introduced in 1907 Jacobi-Perron algorithm, which is the simplest generalization of continued fractions to finite sets of real numbers, the main question of characterising the periodicity is still open. The usual conjecture is that the development of any basis of a real number field by this algorithm is periodic. But we only know some infinite families for which this is true. In this paper we prove that for any real number field there exists a basis for which we have periodicity.

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

Bibliographie

  1. Bernstein, L.: The Jacobi-Perron Algorithm. Its. Theory and Applications. Lecture Notes Math.207. Berlin-Heidelberg-New York: Springer. 1971.

    Google Scholar 

  2. Brentjes, A. I.: Multidimensional continued fraction algorithm. Studieweek Getaltheorie en computers. Amsterdam. 1980. pp. 207–237.

  3. Buchman, J.: Zahlengeometrische Kettenbruchalgorithmen zur Einheitenberechnung. These. Köln 1982.

  4. Bouhamza, M.: Algorithme de Jacobi-Perron dans les corps de nombres de degré 4. Acta Arithmetica (à paraître).

  5. Dubois, E.: Approximations diophantiennes simultanées. Thèse. Paris 1980.

  6. Dubois, E., Paysant-Le Roux, R.: Développements périodiques par l'algorithme de Jacobi-Perron et nombre de Pisot. C. R. Acad. Sci. Paris272, 649–652 (1971).

    Google Scholar 

  7. —: Algorithme de Jacobi-Perron dans les extensions cubiques. C. R. Acad. Sci. Paris280, 183–186 (1975).

    Google Scholar 

  8. Levesque, C.: A class of periodic Jacobi-Perron algorithms in pure algebraic number fields of degreen≥3. Manusc. Math.22, 235–269 (1977).

    Google Scholar 

  9. Mignotte, M.: Une propriété des nombres de Pisot. C. R. A. S. (à paraître).

  10. Perron, O.: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Annalen64, 1–76 (1907).

    Google Scholar 

  11. Pisot, C.: Séminaire de Mathématiques supérieures. Quelques aspects de la théorie des entiers algébriques. Les Presses de l'Université de Montréal: 1963.

  12. Smyth, C. J.: Advanced problems and solutions. Amer. Math. Monthly82, 86 (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Caen, Esplanade de la Paix, F-14032, Caen, France

    R. Paysant-Le Roux & E. Dubois

Authors
  1. R. Paysant-Le Roux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Dubois
    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

Roux, R.PL., Dubois, E. Une application des nombres de Pisot à l'algorithme de Jacobi-Perron. Monatshefte für Mathematik 98, 145–155 (1984). https://doi.org/10.1007/BF01637281

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01637281

Search

Navigation