Lower bounds for fundamental units of real quadratic fields

Abstract

Let $d$ be a square-free positive integer and $l(d)$ be the period length of the simple continued fraction expansion of $\omega_d$, where $\omega_d $ is integral basis of $\mathbb{Z}[\sqrt{d}]$. Let $\varepsilon_d = (t_d+u_d\sqrt{d})/2$ $(>1)$ be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$. In this paper new lower bounds for $\varepsilon_d$, $t_d$, and $u_d$ are described in terms of $l(d)$. The lower bounds of $\varepsilon_d$ are sharper than the known bounds and those of $t_d$ and $u_d$ have been yet unknown. In order to show the strength of the method of the proof, some interesting examples of $d$ are given for which $\varepsilon_d$ and Yokoi's $d$-invariants are determined explicitly in relation to continued fractions of the form $[a_0, \overline{1, \dots ,1, a_{l(d)}}\mkern1.5mu]$.