Continued fractions and certain real quadratic fields of minimal type. (English) Zbl 1151.11057

In this paper the notion of a real quadratic field \(\mathbb Q(\sqrt{d})\) of minimal type that is defined in terms of the length of the period of the continued fraction of \(\sqrt{d}\) or \(\frac{\sqrt{d}+1}{2}\) (taken appropriately) is introduced. The definition while somewhwat technical to be mentioned here, basically only involves quantities arising from the continued fraction mentioned above. If \(\varepsilon=\frac{t+u\sqrt{d}}{2}\) is the fundamental unit, then \(m_d=[\frac{u^2}{t}]\) is called the Yokoi invariant.
The authors show that for given positive integers \(h\) and \(m\), there exist infinitely many real quadratic fields \(\mathbb Q(\sqrt{d})\) with minimal period \(4\) such that the class number \(h_d>h\) and \(m_d=m\) where \(m\) satisfies certain conditions. The main constructional tool used is an improvement of a result by Friesen and Halter-Koch, where an integer \(d\) is constructed so as to have a given continued fraction expansion.
As a consequence the authors prove that the Yokoi invariant of a real quadratic field that is not of minimal type is at most \(3\). It follows that the fundamental unit is small and hence the class number is large. A list of \(51\) real quadratic fields with class number \(1\) that are not of minimal type, with one possible exception is presented. There are ample illustrative examples to make the concepts clear.


11R29 Class numbers, class groups, discriminants
11A55 Continued fractions
11R11 Quadratic extensions
11R27 Units and factorization
Full Text: DOI


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