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Voronoï-algorithm expansion of two families with period length going to infinity. (English) Zbl 0858.11070
G. Rhin and this reviewer previously exhibited two infinite families of periodic Jacobi algorithms with period length going to infinity and involving the field $$\mathbb{Q}(\omega)$$, where $$\omega$$ is a root of $$f(x)$$ (resp. $$g(x))$$, with $$f(x)= x^3-c^mx^2- (c-1) x-c^m$$ (resp. $$g(x) = x^3- (c^m+c-1) x^2- (c^m-1) x-c^m)$$. In a tour de force, the author succeeds in exhibiting periodic Voronoï algorithm expansions (involving the same cubic fields) with period length going to infinity and certainly shows that she masters the subject. Along the same lines and related again to the same cubic fields, it may be worth mentioning the results of E. Dubois and A. Farhane [Util. Math. 47, 97-115 (1995; Zbl 0837.11006)] and J. Kühner [J. Number Theory 53, 1-12 (1995; Zbl 0837.11005)].

##### MSC:
 11Y65 Continued fraction calculations (number-theoretic aspects) 11A55 Continued fractions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants
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