A computational technique for determining the fundamental unit in explicit types of real quadratic number fields

Özer, Özen; Salem, Abdel Badeh M.

International Journal of Advanced and Applied Sciences

Int. j. adv. appl. sci.

EISSN: 2313-3724

Print ISSN: 2313-626X

Volume 4, Issue 2 (February 2017), Pages: 22-27

Title: A computational technique for determining the fundamental unit in explicit types of real quadratic number fields

Author(s): Özen Özer ^1,*, Abdel Badeh M. Salem ²

Affiliation(s):

¹Department of Mathematics, Faculty of Science and Arts, Kırklareli University, 39100, Kırklareli, Turkey
²Faculty of Computer and Informatic Science, Ain Shams University, Cairo, Egypt

https://doi.org/10.21833/ijaas.2017.02.004

Full Text - PDF XML

Abstract:

In real quadratic number field Q(Sqrt (d)), integral basis element is denoted by w_d= [a_0; a₁ to a_l] for the period length l(d). The fundamental unit of real quadratic number field is also denoted by Epsilon_d. The Unit Theorem for real quadratic fields says that every unit in the integer ring of a quadratic field is generated by the fundamental unit. Also, regulator in real quadratic cryptography is outstanding. We have seen that the regulator R plays the role of a group order. The regulator problem is to find an integer R' satisfies |R'-R|<1 where R' is an approximation of with any given precision can be computed in polynomial time for discriminant. However, some of the fundamental units can not be calculated by computer programme in short time because of the big numbers or long calculations of usual algorithm. This is also the main problem from the computing/informatics point of view. So, determining of the fundamental units is of great importance. In this paper, we construct a theorem to determine the some certain real quadratic fields Q(Sqrt(d)) having specific form of continued fraction expansion of w_d where d is a square-free integer. We also present the general context and obtain new certain parametric representation of fundamental unit for such types of fields.By specialization, we get a fix on Yokoi’s invariants and support all results with tables.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Quadratic fields, Continued fractions, Fundamental units, Yokoi’s invariants

Article History: Received 24 October 2016, Received in revised form 10 January 2017, Accepted 10 January 2017

Digital Object Identifier:

https://doi.org/10.21833/ijaas.2017.02.004

Citation:

Özer Ö and Salem ABM (2017). A computational technique for determining the fundamental unit in explicit types of real quadratic number fields. International Journal of Advanced and Applied Sciences, 4(2): 22-27

http://www.science-gate.com/IJAAS/V4I2/Özer.html

References:

Badziahin D and Shallit J (2016). An unusual continued fraction. Proceedings of the American Mathematical Society, 144(5): 1887-1896. https://doi.org/10.1090/proc/12848

Benamar H, Chandoul A, and Mkaouar M (2015). On the continued fraction expansion of fixed period in finite fields. Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques, 58(4): 704-712. https://doi.org/10.4153/CMB-2015-055-9

Buchmann J (2004). Introduction to cryptography. 2nd Edition, Springer-Verlag, New York, USA. https://doi.org/10.1007/978-1-4419-9003-7

Clemens LE, Merill KD, and Roeder DW (1995). Continued fractions and series. Journal of Number Theory, 54(2): 309-317. https://doi.org/10.1006/jnth.1995.1121

Elezovi´ N (1997). A note on continued fractions of quadratic irrationals. Mathematical Communications, 2(1): 27-33.

Kawamoto F and Tomita K (2008). Continued fractions and certain real quadratic fields of minimal type. Journal of the Mathematical Society of Japan, 60(3): 865-903. https://doi.org/10.2969/jmsj/06030865

Louboutin S (1988). Continued fractions and real quadratic fields. Journal of Number Theory, 30(2): 167-176. https://doi.org/10.1016/0022-314X(88)90015-7

Mollin RA (1996). Quadratics. CRC Press, Boca Rato, FL.

Olds CD (1963). Continued fractions. Random House, New York, USA

Özer Ö (2016a). Notes on especial continued fraction expansions and real quadratic number fields. Kirklareli University Journal of Engineering and Science, 2(1): 74-89.

Özer Ö (2016b). On real quadratic number fields related with specific type of continued fractions. Journal of Analysis and Number Theory, 4(2): 85-90. https://doi.org/10.18576/jant/040201

Perron O (1950). Die Lehre von den Kettenbrichen. Chelsea, reprint from teubner leipzig, New York, USA. PMid:14785649

Sasaki R (1986). A characterization of certain real quadratic fields. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 62(3): 97-100. https://doi.org/10.3792/pjaa.62.97

Sierpinski W (1964). Elementary theory of numbers. Monografi Matematyczne, Warsaw. PMid:14200537

Tomita K (1995). Explicit representation of fundamental units of some quadratic fields. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 71(2): 41-43. https://doi.org/10.3792/pjaa.71.41

Tomita K and Yamamuro K (2002). Lower bounds for fundamental units of real quadratic fields. Nagoya Mathematical Journal, 166: 29-37. https://doi.org/10.1017/S0027763000008230

Williams KS and Buck N (1994). Comparison of the lengths of the continued fractions of √ (D) and fraction 1/2 (1+√ (D)). Proceedings of the American Mathematical Society, 120(4): 995-1002.

Yokoi H (1990). The fundamental unit and class number one problem of real quadratic fields with prime discriminant. Nagoya Mathematical Journal, 120: 51-59. https://doi.org/10.1017/S0027763000003238

Yokoi H (1991). The fundamental unit and bounds for class numbers of real quadratic fields. Nagoya Mathematical Journal, 124: 181-197. https://doi.org/10.1017/S0027763000003834

Yokoi H (1993a). A note on class number one problem for real quadratic fields. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 69(1): 22-26. https://doi.org/10.3792/pjaa.69.22

Yokoi H (1993b). New invariants and class number problem in real quadratic fields. Nagoya Mathematical Journal, 132: 175-197. https://doi.org/10.1017/S0027763000004700

Zhang Z and Yue Q (2014). Fundamental units of real quadratic fields of odd class number. Journal of Number Theory, 137: 122-129. https://doi.org/10.1016/j.jnt.2013.10.019