Fundamental units for real quadratic fields determined by continued fraction conditions

Abstract

The aim of this paper is to obtain the real quadratic fields $\mathbb{Q}\left(\sqrt{d}\right)$ including $$\omega_d=\left[a_0;;\overline{\underbrace{\gamma,\gamma,\dots,\gamma}_{l-1},a_l}\right]$$ where $l=l\left(d\right)$ is the period length and $\gamma$ is a positive odd integer. Moreover, we have considered a new perspective to determine the fundamental units $\epsilon_d$ and got important results on Yokoi's invariants $n_d$ and $m_d$ [since they satisfy necessary and sufficient conditions related to Ankeny-Artin-Chowla conjecture (A.A.C.C), give bounds for fundamental units and so on...] for such types of fields.

Keywords

• continued fraction expansion
• real quadratic number field
• fundamental unit
• special sequence
• yokoi’s invariants