Journal of the Mathematical Society of Japan

Continued fractions and certain real quadratic fields of minimal type

Fuminori KAWAMOTO and Koshi TOMITA

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Abstract

The main purpose of this article is to introduce the notion of real quadratic fields of minimal type in terms of continued fractions with period . We show that fundamental units of real quadratic fields that are not of minimal type are relatively small. So, we see by a theorem of Siegel that such fields have relatively large class numbers. Also, we show that there exist exactly 51 real quadratic fields of class number 1 that are not of minimal type, with one more possible exception. All such fields are listed in the table of Section 8.2. Therefore we study real quadratic fields with period of minimal type in order to find real quadratic fields of class number 1 , and first examine the case where 4 . In particular we obtain a result on Yokoi invariants m d and class numbers h d of real quadratic fields Q( d ) with period 4 of minimal type.

Article information

Source
J. Math. Soc. Japan, Volume 60, Number 3 (2008), 865-903.

Dates
First available in Project Euclid: 4 August 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1217884495

Digital Object Identifier
doi:10.2969/jmsj/06030865

Mathematical Reviews number (MathSciNet)
MR2440416

Zentralblatt MATH identifier
1151.11057

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11R11: Quadratic extensions 11R27: Units and factorization

Keywords
continued fractions real quadratic fields fundamental units class numbers

Citation

KAWAMOTO, Fuminori; TOMITA, Koshi. Continued fractions and certain real quadratic fields of minimal type. J. Math. Soc. Japan 60 (2008), no. 3, 865--903. doi:10.2969/jmsj/06030865. https://projecteuclid.org/euclid.jmsj/1217884495


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