Continued fractions and certain real quadratic fields of minimal type

概要

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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 $¥ell$ ≤ 4. In particular we obtain a result on Yokoi invariants md and class numbers hd of real quadratic fields Q($¥sqrt{d}$) with period 4 of minimal type.
2008-07-01