manuscripta mathematica

, Volume 22, Issue 3, pp 235–269 | Cite as

A class of periodic Jacobi-Perron Algorithms in pure algebraic number fields of degree n≥3

  • Claude Levesque


The author obtains the periodicity of the Jacobi-Perron algorithm of (θ,...,θn−1) where θn=(Dn−d)/Vn, with D, d,V ∈ N*=1,2,..., d|D, D and d congruent to +1(mod Vn−1) and D≥(n−1)d(V+1)/2+1. The case V=1 has been studied by L. Bernstein and the proof for arbitrary V follows exactly the same pattern. Secondary results are then obtained from the main theorem.


Secondary Result Number Theory Algebraic Geometry Topological Group Number Field 
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  1. [1]
    L. Bernstein, Lecture Notes in Mathematics207. The Jacobi-Perron Algorithm, Its Theory and Application. Berlin-Heidelberg-New York (1971).Google Scholar
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    L. Bernstein, Representation of\(\sqrt[n]{{D^n - d}}\) as a Periodic Continued Fraction by Jacobi's Algorithm, Math. Nachr.29, 179–200, (1965).Google Scholar
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    L. Bernstein, New Infinite Classes of Periodic Jacobi-Perron Algorithms, Pac. J. of Math.,16, 1–31, (1965).Google Scholar
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    L. Bernstein, Units from Periodic Jacobi-Perron-Algorithms in Algebraic Number Fields of degree n>2, Manuscripta Math.14, 249–261, (1974).Google Scholar
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    L. Bernstein, Units and Periodic Jacobi-Perron Algorithms in Real Algebraic Number Fields of degree 3, Trans. Am. Math. Soc.212, 295–306, (1975).Google Scholar
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    C. Levesque, A Class of Fundamental Units and some Classes of Jacobi-Perron Algorithms in Pure Cubic Fields, Pac. J. of Math., accepted for publication.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Claude Levesque
    • 1
  1. 1.Département de MathématiquesUniversité LavalCanada

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