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Two families of periodic Jacobi algorithms with period lengths going to infinity. (English) Zbl 0723.11032
Let \(c\geq 2\) and \(m\geq 1\) be fixed integers. The authors establish that the Jacobi-Perron expansion of the vector \((x^ 2-(c^ m+c-1)x,x)\) is purely periodic of period length \(4m+1\), where x is selected to satisfy \(c^ m+c-1<x<c^ m+c.\) The key to their solution is to show that \(g(x)=x^ 3-(c^ m+c-1)x^ 2-(c^ m-1)x-c^ m\) has only one real root and that root x satisfies the previous inequalities. By a similar consideration it is also established that the Jacobi-Perron expansion of the vector \((x^ 2-c^ mx,x)\) is purely periodic, this time with period length \(3m+1\).

MSC:
11J70 Continued fractions and generalizations
11A67 Other number representations
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[1] Bernstein, L., The Jacobi-Perron algorithm, its theory and application, () · Zbl 0213.05201
[2] Dubois, E.; Paysant-Le Roux, R., Développement périodique par l’algorithme de Jacobi-Perron et nombre de Pisot-vijayaraghavan, C. R. acad. sci. Paris Sér. I math., 272, 649-652, (1971) · Zbl 0207.05402
[3] Dubois, E.; Paysant-Le Roux, R., Algorithmes de Jacobi-Perron dans LES extensions cubiques, C. R. acad. sci. Paris Sér. I math., 280, 183-186, (1975) · Zbl 0297.12002
[4] Paysant-Le Roux, R.; Dubois, E., De nouvelles classes d’entiers pour lesquelles on connaît des développements périodiques par l’algorithme de Jacobi-Perron dans ℚ(m^{1/(n+1)}), C. R. acad. sci. Paris Sér. I math., 280, 57-59, (1975) · Zbl 0297.12001
[5] Levesque, C., A class of fundamental units and some classes of Jacobi-Perron algorithms in pure cubic fields, Pacific J. math., 81, 447-466, (1979) · Zbl 0427.12003
[6] Perron, O., Grundlagen für eine theorie des jacobischen kettenbruchalgorithmus, Math. ann., 64, 1-76, (1907) · JFM 38.0262.01
[7] Perron, O., Der Jacobi’sche kettenalgorithmus in einem kubischen zahlenkörper, München ak. sb., 13-49, (1971) · Zbl 0249.10022
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