Une application des nombres de Pisot à l’algorithme de Jacobi-Perron. (French) Zbl 0543.10023

Since O. Perron introduced in 1907 the Jacobi-Perron algorithm, which is the simplest generalization of continued fractions to finite sets of real numbers, the main question of characterizing 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. For this we use the property ”The conjugates of a Pisot number are multiplicatively independent”. We also give some numerical examples.


11J70 Continued fractions and generalizations
11A55 Continued fractions
11-04 Software, source code, etc. for problems pertaining to number theory
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Full Text: DOI EuDML


[1] Bernstein, L.: The Jacobi-Perron Algorithm. Its. Theory and Applications. Lecture Notes Math.207. Berlin-Heidelberg-New York: Springer. 1971. · Zbl 0213.05201
[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). · Zbl 0551.12001
[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). · Zbl 0207.05402
[7] –: Algorithme de Jacobi-Perron dans les extensions cubiques. C. R. Acad. Sci. Paris280, 183–186 (1975). · Zbl 0297.12002
[8] Levesque, C.: A class of periodic Jacobi-Perron algorithms in pure algebraic number fields of degreen. Manusc. Math.22, 235–269 (1977). · Zbl 0375.10018
[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). · JFM 38.0262.01
[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). · Zbl 0326.53055
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