## 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.

### MSC:

 11R29 Class numbers, class groups, discriminants 11A55 Continued fractions 11R11 Quadratic extensions 11R27 Units and factorization
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### References:

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