Bulletin of the
Korean Mathematical Society BKMS

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 $S(l; a_1, \ldots, a_{l-1}):=\{d \in \mathbb Z\,|\, d>0, \,\sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]\}$. 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.

Keywords: units, real quadratic fields

MSC numbers: 11R11, 11R27