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.