Abstract
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.
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References
L. Bernstein, Lecture Notes in Mathematics207. The Jacobi-Perron Algorithm, Its Theory and Application. Berlin-Heidelberg-New York (1971).
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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).
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.
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Levesque, C. A class of periodic Jacobi-Perron Algorithms in pure algebraic number fields of degree n≥3. Manuscripta Math 22, 235–269 (1977). https://doi.org/10.1007/BF01172666
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DOI: https://doi.org/10.1007/BF01172666