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Lower bounds for fundamental units of real quadratic fields

Published online by Cambridge University Press:  22 January 2016

Koshi Tomita  and
Kouji Yamamuro
Koshi Tomita
Affiliation:
Department of Mathematics, Meijo University, Shiogamaguchi 1-501, Tenpaku-ku, Nagoya, 468-8502, Japan, tomita@ccmfs.meijo-u.ac.jp
Kouji Yamamuro
Affiliation:
Department of Liberal Arts, Aichi Konan College, Omatsubara 172, Takaya-cho, Konan-shi, Aichi, 483-8086, Japan, yamamuro@math.nagoya-u.ac.jp
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Abstract

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Let d be a square-free positive integer and l(d) be the period length of the simple continued fraction expansion of ωd, where ωd is integral basis of ℤ[]. Let εd = (td + ud)/2 (> 1) be the fundamental unit of the real quadratic field ℚ(). In this paper new lower bounds for εd, td, and ud are described in terms of l(d). The lower bounds of εd are sharper than the known bounds and those of td and ud have been yet unknown. In order to show the strength of the method of the proof, some interesting examples of d are given for which εd and Yokoi’s d-invariants are determined explicitly in relation to continued fractions of the form .

Keywords

11R1111R27
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 166 , June 2002 , pp. 29 - 37
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

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