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，即每个周期的最小元给出一个类数为1的最小型实数二次域。(2)我们研究了在每个偶周期内构造这样一个域，然后Kawamoto-Kishi-Tomita（2014）给出了一种构造极小型偶周期正整数的方法。(3)利用这种构造，我们引入了“极大端（ELE）”的概念，用于初级对称部分，“前ELE型”的概念用于有限弦。然后给出了构造ELE型的主对称部分的一种方法。(4)进一步引入了pre-ELE型有限串的“生长分解”概念，给出了构造pre-ELE型有限串的一种方法。作为副产物，我们证明了在每一个偶数且大于等于6的周期内存在无穷多个极小型实数二次域。

(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.