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，即每个周期的极小元给出一个类数为1的实二次域的极小类型。(2)研究了在每个偶数周期内构造这样一个域，然后Kawamoto-Kishi-Tomita(2014)给出了构造极小偶数周期的正整数的方法。(3)利用这种结构，我们引入了主对称部分的“极大端”和有限弦的“预大端”的概念。然后给出了一种构造ELE型初等对称部分的方法。(4)此外，我们还引入了Pre-EL型有限串的“增长分解”的概念，并给出了构造Pre-EL型有限串的方法。作为一个副产品，我们证明了在每个偶数且大于或等于6的周期内，存在无穷多个极小型实二次场。

(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.