Continued fractions with even period and an infinite family of real quadratic fields of minimal type

Abstract

In a previous paper [4], we introduced the notion of real quadratic fields with period $l$ of minimal type in terms of continued fractions. As a consequence, we have to examine a construction of real quadratic fields with period $\ge 5$ of minimal type in order to find many real quadratic fields of class number 1. When $l \ge 4$, it appears that there exist infinitely many real quadratic fields with period $l$ of minimal type. Indeed, we provided an infinitude of real quadratic fields with period 4 of minimal type in [4]. In this paper, we construct an infinite family of real quadratic fields with large even period of minimal type whose class number is greater than any given positive integer, and whose Yokoi invariant is greater than any given positive integer.