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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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참고문헌 (8)
참고문헌 신청
A. J. Van Der Poorten / 2001 / Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000 / Math. Comp 70 (235) : 1311 ~ 1328
A. J. Van Der Poorten / 2003 / Corrigenda and addition to \Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000 / Math. Comp 7
B. D. Beach / 1971 / Some computer results on units in qua- dratic and cubic fields / Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress : 609 ~ 6
J. Mc Laughlin / 2003 / Multi-variable polynomial solutions to Pell's equation and fundamental units in real quadratic fields / Pacific J. Math 210 (2) : 335 ~ 349
K. Tomita / 1997 / Explicit representation of fundamental units of some real quadratic fields. II / J. Number Theory 63 (2) : 275 ~ 285
A. J. Van Der Poorten / 2003 / Corrigenda and addition to \Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000 / Math. Comp 7
B. D. Beach / 1971 / Some computer results on units in qua- dratic and cubic fields / Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress : 609 ~ 6
J. Mc Laughlin / 2003 / Multi-variable polynomial solutions to Pell's equation and fundamental units in real quadratic fields / Pacific J. Math 210 (2) : 335 ~ 349
K. Tomita / 1997 / Explicit representation of fundamental units of some real quadratic fields. II / J. Number Theory 63 (2) : 275 ~ 285
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