A NOTE ON UNITS OF REAL QUADRATIC FIELDS

For a positive square-free integer $d$, let $t_d$ and $u_d$ be positive integers such that ${\epsilon}_d=\frac{t_d+u_d{\sqrt{d}}}{\sigma}$ is the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$, where ${\sigma}=2$ if $d{\equiv}1$ (mod 4) and ${\sigma}=1$ otherwise For a given positive integer $l$ and a palindromic sequence of positive integers $a_1$, ${\ldots}$, $a_{l-1}$, we define the set $S(l;a_1,{\ldots},a_{l-1})$ := {$d{\in}\mathbb{Z}|d$ > 0, $\sqrt{d}=[a_0,\overline{a_1,{\ldots},2a_0}]$}. We prove that $u_d$ < $d$ for all square-free integer $d{\in}S(l;a_1,{\ldots},a_{l-1})$ with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.