A NOTE ON UNITS OF REAL QUADRATIC FIELDS

Abstract

For a positive square-free integer <TEX>$d$</TEX>, let <TEX>$t_d$</TEX> and <TEX>$u_d$</TEX> be positive integers such that <TEX>${\epsilon}_d=\frac{t_d+u_d{\sqrt{d}}}{\sigma}$</TEX> is the fundamental unit of the real quadratic field <TEX>$\mathbb{Q}(\sqrt{d})$</TEX>, where <TEX>${\sigma}=2$</TEX> if <TEX>$d{\equiv}1$</TEX> (mod 4) and <TEX>${\sigma}=1$</TEX> otherwise For a given positive integer <TEX>$l$</TEX> and a palindromic sequence of positive integers <TEX>$a_1$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$a_{l-1}$</TEX>, we define the set <TEX>$S(l;a_1,{\ldots},a_{l-1})$</TEX> := {<TEX>$d{\in}\mathbb{Z}|d$</TEX> > 0, <TEX>$\sqrt{d}=[a_0,\overline{a_1,{\ldots},2a_0}]$</TEX>}. We prove that <TEX>$u_d$</TEX> < <TEX>$d$</TEX> for all square-free integer <TEX>$d{\in}S(l;a_1,{\ldots},a_{l-1})$</TEX> with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.

Locations

• Bulletin of the Korean Mathematical Society - View - PDF