A construction of real quadratic fields of minimal type and primary symmetric parts of ELE type (Algebraic Number Theory and Related Topics 2014)

Abstract

This article is an announcement of our recent papers [6] and [5]. For a non-square positive integer d with 4 ∤ d, put ω(d) := (1 + √d)=2 if d is congruent to 1 modulo 4 and otherwise ω(d) := √d. Let a1, a2, …, aℓ-1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the string a1, a2, …, a[ℓ/2] is the primary symmetric part of the simple continued fraction expansion of ω(d). The main purposes of this article are to introduce notions of "ELE type" and "pre-ELE type" for a finite string of positive integers, and to study properties for a non-square positive integer d such that the primary symmetric part of the simple continued fraction expansion of √d with even period is of ELE type.