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 $Q(\sqrt{d})$ with period $4$ of minimal type.