Bulletin of the Korean Mathematical Society (대한수학회보)

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A NOTE ON UNITS OF REAL QUADRATIC FIELDS

  • Byeon, Dong-Ho (Department of Mathematics Seoul National University) ;
  • Lee, Sang-Yoon (Department of Mathematics Seoul National University)
  • Received : 2011.04.05
  • Published : 2012.07.31
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Abstract

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.

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References

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  2. REAL QUADRATIC FUNCTION FIELDS OF MINIMAL TYPE vol.28, pp.4, 2013, https://doi.org/10.4134/CKMS.2013.28.4.735
  3. Fundamental units and consecutive squarefull numbers vol.13, pp.01, 2017, https://doi.org/10.1142/S1793042117500142