Application of continued fractions to real quadratic fields

(1)Kawamoto-Tomita（2012）利用数值数据提出了一个猜想：每个周期的极小元给出一个类数为1的极小型真实的二次域。(2)我们研究在每个偶数周期中构造这样的域，然后Kawamoto-Kishi-Tomita（2014）给出了一种构造具有极小型偶数周期的正整数的方法。(3)通过使用这种结构，我们引入了“极大端（ELE）”的主要对称部分和“前ELE类型”的有限字符串的概念。然后给出了一种构造ELE型准素对称部分的方法。(4)此外，我们还对pre-ELE型有限串引入了“增长分解”的概念，从而给出了一种构造pre-ELE型有限串的方法。作为一个副产品，我们证明了存在无穷多个真实的二次域的极小型在每个周期是偶数和大于或等于6。

(1) Kawamoto-Tomita(2012) posed a conjecture by using numerical datas that the minimal element of each period gives a real quadratic field with class number 1 of minimal type. (2) We examine to construct such a field in each even period, and then Kawamoto-Kishi-Tomita(2014) gave a way of constructing positive integers with even period of minimal type. (3) By using this construction, we introduce notions of ``extremely large end (ELE)'' for a primary symmetric part and of ``pre-ELE type'' for a finite string. Then we give a way of constructing primary symmetric parts of ELE type. (4) Moreover, we introduce a notion of ``a growth decomposition'' for a finite string of pre-ELE type, and then give a way of constructing finite strings of pre-ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields of minimal type in each period which is even and greater than or equal to 6.