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

Document Type: Research Paper

Authors

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

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.

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