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Comparison of the lengths of the continued fractions of $ \sqrt D$ and $ \frac12(1+\sqrt D)$

Authors: Kenneth S. Williams and Nicholas Buck
Journal: Proc. Amer. Math. Soc. 120 (1994), 995-1002
MSC: Primary 11J70
MathSciNet review: 1169053
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Abstract: Let $ D$ denote a positive nonsquare integer such that $ D \equiv 1\,(\bmod 4)$. Let $ l(\sqrt D )$ (resp. $ l(\tfrac{1} {2}(1 + \sqrt D ))$) denote the length of the period of the continued fraction expansion of $ \sqrt D $ (resp. $ \tfrac{1} {2}(1 + \sqrt D ))$). Recently Ishii, Kaplan, and Williams (On Eisenstein's problem, Acta Arith. 54 (1990), 323-345) established inequalities between $ l(\sqrt D )$ and $ l(\tfrac{1} {2}(1 + \sqrt D ))$. In this note it is shown that these inequalities are best possible in a strong sense.

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Keywords: Continued fractions
Article copyright: © Copyright 1994 American Mathematical Society

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