A NOTE ON UNITS OF REAL QUADRATIC FIELDS

Title & Authors
A NOTE ON UNITS OF REAL QUADRATIC FIELDS
Byeon, Dong-Ho; Lee, Sang-Yoon;

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

대한수학회논문집, 2013. vol.28. 4, pp.735-740
2.
ON CONTINUED FRACTIONS, FUNDAMENTAL UNITS AND CLASS NUMBERS OF REAL QUADRATIC FUNCTION FIELDS,;

충청수학회지, 2014. vol.27. 2, pp.183-203
1.
ON CONTINUED FRACTIONS, FUNDAMENTAL UNITS AND CLASS NUMBERS OF REAL QUADRATIC FUNCTION FIELDS, Journal of the Chungcheong Mathematical Society, 2014, 27, 2, 183
2.
REAL QUADRATIC FUNCTION FIELDS OF MINIMAL TYPE, Communications of the Korean Mathematical Society, 2013, 28, 4, 735
3.
Fundamental units and consecutive squarefull numbers, International Journal of Number Theory, 2017, 13, 01, 243
References
1.
B. D. Beach, H. C. Williams, and C. R. Zarnke, Some computer results on units in quadratic and cubic fields, Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971), pp. 609-648, Lake-head Univ., Thunder Bay, Ont., 1971.

2.
R. Hashimoto, Ankeny-Artin-Chowla conjecture and continued fraction, J. Number Theory 90 (2001), no. 1, 143-153.

3.
J. Mc Laughlin, Multi-variable polynomial solutions to Pell's equation and fundamental units in real quadratic fields, Pacific J. Math. 210 (2003), no. 2, 335-349.

4.
R. A. Mollin, Quadratics, CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, 1996.

5.
R. A. Mollin and P. G. Walsh, A note on powerful numbers, quadratic fields and the Pellian, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 2, 109-114.

6.
A. J. Van Der Poorten, H. J. J. te Riele, and H. C. Williams, Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000, Math. Comp. 70 (2001), no. 235, 1311-1328.

7.
A. J. Van Der Poorten, H. J. J. te Riele, and H. C. Williams, Corrigenda and addition to \Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000", Math. Comp. 72 (2003), no. 241, 521-523.

8.
K. Tomita, Explicit representation of fundamental units of some real quadratic fields. II, J. Number Theory 63 (1997), no. 2, 275-285.