On the real quadratic fields with certain continued fraction expansions and fundamental units

Ozer, Ozen; Khammash, Ahmed

doi:10.22075/ijnaa.2017.1610.1420

On the real quadratic fields with certain continued fraction expansions and fundamental units

Document Type : Research Paper

Authors

Ozen Ozer ¹

Ahmed Khammash ²

¹ Department of Mathematics, Faculty of Science and Arts, Ki rklareli University, 39000-Ki rklareli, Turkey

² Department of Mathematics, Al-Qura University, Makkah,21955, Saudi Arabia

10.22075/ijnaa.2017.1610.1420

Abstract

The purpose of this paper is to investigate the real quadratic number fields $Q(\sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element where $d\equiv 2,3( mod 4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit
$$\epsilon _{d}=\left(t_d+u_d\sqrt{d}\right)\ 2\left.\right > 1$$
and $n_d$ and $m_d$ Yokoi's $d$-invariants by reference to continued fraction expansion of integral basis element where $\ell \left({d}\right)$ is a period length. Moreover, we mention class number for such fields. Also, we give some numerical results concluded in the tables.

Keywords

Quadratic Field

Fundamental Unit

Continued Fraction Expansion

Class Number