Continued fractions and certain real quadratic fields of minimal type

被引:12
|
作者
Kawamoto, Fuminori [1 ]
Tomita, Koshi [2 ]
机构
[1] Gakushuin Univ, Fac Sci, Dept Math, Toshima Ku, Tokyo 1718588, Japan
[2] Meijo Univ, Dept Math, Tenpaku Ku, Nagoya, Aichi 4688502, Japan
关键词
continued fractions; real quadratic fields; fundamental units; class numbers;
D O I
10.2969/jmsj/06030865
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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 e. 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 9 of minimal type in order to find real quadratic fields of class number 1, and first examine the case where l <= 4. In particular we obtain a result on Yokoi invariants m(d) and class numbers h(d) of real quadratic fields Q(root d) with period 4 of minimal type.
引用
收藏
页码:865 / 903
页数:39
相关论文
共 50 条