A description of continued fraction expansions of quadratic surds represented by polynomials

JOURNAL OF NUMBER THEORY

卷:107

Article

Mollin, RA ; Goddard, B

关键词: continued fractions;    quadratic irrational;    period length;    fundamental units;   

DOI  :  10.1016/j.jnt.2004.02.004

来源: Elsevier

【 摘 要 】

The principal thrust of this investigation is to provide families of quadratic polynomials {D-k(X) = f(k)(2)X(2) + 2e(k)X + C}(kepsilonN), where e(k)(2) - f(k)(2)C = n (for any given nonzero integer n) satisfying the property that for any X epsilon N, the period length l(k) = e(rootD(k) -(X)) of the simple continued fraction expansion of rootD(k)(X) is constant for fixed k and lim(k-->infinity) l(k) = infinity. This generalizes, and completes, numerous results in the literature, where the primary focus was upon \n\ = 1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part 111, 257-293; Serdica Math. J. 27 (2001) 317-342) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102-108; Internat Math J 2 (2002) 951-956) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47-60). (C) 2004 Elsevier Inc. All rights reserved.

【 授权许可】

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