Bulletin of the
Korean Mathematical Society BKMS

It is well known that the continued fraction expansion of $\sqrt{d}$ has the form $[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$ and $a_1, \ldots, a_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer $l$ and a palindromic sequence of positive integers $a_1, \ldots, a_{l-1}$, we define the set $S(l;a_1,$ $\ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}], \, \textup{where} \, a_0=\lfloor \sqrt{d} \rfloor\}$. In this paper, we completely determine when $S(l;a_1, \ldots, a_{l-1})$ is not empty in the case that $l$ is $4$, $5$, $6$, or $7$. We also give similar results for $(1+\sqrt{d})/2$. For the case that $l$ is $4$, $5$, or $6$, we explicitly describe the fundamental units of the real quadratic field $\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of $\mathbb{Q}(\sqrt{d})$.

Keywords: Continued fractions, quadratic fields, fundamental units

MSC numbers: Primary 11A55; Secondary 11R11, 11R27

Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(the Ministry of Education and MSIT) (2017 R1D1A1B03033560, 2020R1F1A1A01069118).