Bull. Korean Math. Soc. 2012; 49(4): 767-774
Printed July 1, 2012
https://doi.org/10.4134/BKMS.2012.49.4.767
Copyright © The Korean Mathematical Society.
Dongho Byeon and Sangyoon Lee
Seoul National University, Seoul National University
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
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd