## Journal of the Mathematical Society of Japan

### Continued fractions and certain real quadratic fields of minimal type

#### 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 $\ell$. 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 $\ell$ of minimal type in order to find real quadratic fields of class number $1$, and first examine the case where $\ell\le 4$. In particular we obtain a result on Yokoi invariants $m_d$ and class numbers $h_d$ of real quadratic fields $\bm{Q}(\sqrt{d})$ 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

https://projecteuclid.org/euclid.jmsj/1217884495

Digital Object Identifier
doi:10.2969/jmsj/06030865

Mathematical Reviews number (MathSciNet)
MR2440416

Zentralblatt MATH identifier
1151.11057

#### 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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