## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 46, Number 4 (2009), 949-993.

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

Fuminori Kawamoto and Koshi Tomita

#### 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.

#### Article information

**Source**

Osaka J. Math., Volume 46, Number 4 (2009), 949-993.

**Dates**

First available in Project Euclid: 15 December 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1260892836

**Mathematical Reviews number (MathSciNet)**

MR2604917

**Zentralblatt MATH identifier**

1247.11140

**Subjects**

Primary: 11R29: Class numbers, class groups, discriminants

Secondary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11R11: Quadratic extensions 11R27: Units and factorization

#### Citation

Kawamoto, Fuminori; Tomita, Koshi. Continued fractions with even period and an infinite family of real quadratic fields of minimal type. Osaka J. Math. 46 (2009), no. 4, 949--993. https://projecteuclid.org/euclid.ojm/1260892836