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On the fundamental units and the class numbers of real quadratic fields. (English) Zbl 0533.12008
Let h(m) be the class number of the real quadratic field \({\mathbb{Q}}(\sqrt{m})\). When p is a prime number such that \(p=a^ 2+1\), \(a^ 2+4\), \(a^ 2\pm 2\), the sufficient conditions for \(h(p)>1\) were given by several authors. In this paper, we extend these results. First, we show the recurrence formula with rational integers for the continued fraction expansion of quadratic irrationals. We use it to get the sufficient conditions for \(h(p)>1\) with above types of p, from the result that the number of equivalence classes of reduced quadratic irrationals is equal to the class number. We also get the fundamental units of \({\mathbb{Q}}(\sqrt{m})\) for several types of m.

MSC:
11R27 Units and factorization
11R23 Iwasawa theory
11R11 Quadratic extensions
11R80 Totally real fields
11A55 Continued fractions
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