Fundamental units for real quadratic fields determined by continued fraction conditions

Özen Özer; Özen Özer

doi:10.3934/math.2020187

AIMS Mathematics

2020, Volume 5, Issue 4: 2899-2908. doi: 10.3934/math.2020187

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Research article

Fundamental units for real quadratic fields determined by continued fraction conditions

Özen Özer ^,

Department of Mathematics, Faculty of Science and Arts, Kırklareli University, 39100-Kırklareli, Turkey

Received: 31 October 2019 Accepted: 03 March 2020 Published: 19 March 2020
MSC : 11A55, 11R11, 11R27, 11R29, 11Y55, 11K83

The aim of this paper is to obtain the real quadratic fields $\mathbb{Q}\left(\sqrt{d}\right)$ including $\omega_d = \left[a_0;;\overline{\underbrace{\gamma,\gamma,\dots,\gamma}_{l-1},a_l}\right] $ where $l = l\left(d\right)$ is the period length and $\gamma$ is a positive odd integer. Moreover, we have considered a new perspective to determine the fundamental units $\epsilon_d$ and got important results on Yokoi's invariants $n_d$ and $m_d$ [since they satisfy necessary and sufficient conditions related to Ankeny-Artin-Chowla conjecture (A.A.C.C), give bounds for fundamental units and so on...] for such types of fields.
- continued fraction expansion,
- real quadratic number field,
- fundamental unit,
- special sequence,
- Yokoi’s invariants
Citation: Özen Özer. Fundamental units for real quadratic fields determined by continued fraction conditions[J]. AIMS Mathematics, 2020, 5(4): 2899-2908. doi: 10.3934/math.2020187

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Abstract

The aim of this paper is to obtain the real quadratic fields $\mathbb{Q}\left(\sqrt{d}\right)$ including $\omega_d = \left[a_0;;\overline{\underbrace{\gamma,\gamma,\dots,\gamma}_{l-1},a_l}\right] $ where $l = l\left(d\right)$ is the period length and $\gamma$ is a positive odd integer. Moreover, we have considered a new perspective to determine the fundamental units $\epsilon_d$ and got important results on Yokoi's invariants $n_d$ and $m_d$ [since they satisfy necessary and sufficient conditions related to Ankeny-Artin-Chowla conjecture (A.A.C.C), give bounds for fundamental units and so on...] for such types of fields.

References

[1]	J. Buchmann, T. Tagaki, U. Vollmer, Number field cryptography, Fields Institute Communication, 41 (2004), 111-121.
[2]	D. Badziahin, J. Shallit, An unusual continued fraction, Proc. Amer. Math. Soc., 144 (2016), 1887-1896. doi: 10.1090/proc/12848
[3]	H. Benamar, A. Chandoul, M. Mkaouar, On the continued fraction expansion of fixed period in finite fields, Canada Math. Bull., 58 (2015), 704-712. doi: 10.4153/CMB-2015-055-9
[4]	K. Chakraborty, S. Kanemitsu, T. Kuzumaki, On the class number formula of certain real quadratic fields, Hardy Ramanujan J., 36 (2013), 1-7. doi: 10.46298/hrj.2013.178
[5]	L. E. Clemens, K. D. Merill, D. W. Roeder, Continued fractions and series, J. Number Theory, 54 (1995), 309-317. doi: 10.1006/jnth.1995.1121
[6]	N. Elezovic, A note on continued fractions of quadratic irrationals, Math. Commun., 2 (1997), 27-33.
[7]	C. Friesen, On continued fraction of given period, Proc. Amer. Math. Soc., 10 (1988), 9-14. doi: 10.1090/S0002-9939-1988-0938635-4
[8]	F. Halter-Koch, Continued fractions of given symmetric period, Fibonacci Quart., 29 (1991), 298-303.
[9]	P. Jeangho, Notes on the quadratic integer and real quadratic number field, Osaka J. Math., 53 (2016), 983-1002.
[10]	F. Kawamoto, K. Tomita, Continued fraction and certain real quadratic fields of minimal type, J. Math. Soc., 60 (2008), 865-903. doi: 10.2969/jmsj/06030865
[11]	J. M. Kim, J. Ryu, On the class number and fundamental unit of the quadratic field $k=\mathbb{Q}\left(\sqrt{p.q}\right)$, Bull. Aus. Math. Soc., 85 (2012), 359-370.
[12]	S. Louboutin, Continued fraction and real quadratic fields, J. Number Theory, 30 (1988), 167-176. doi: 10.1016/0022-314X(88)90015-7
[13]	J. McLaughlin, Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields, J. London Math. Soc., 67 (2003), 16-28. doi: 10.1112/S002461070200371X
[14]	R. A. Mollin, Quadratics, 3^rd Ed, CRC Press, Boca Raton, FL, 1996.
[15]	Ö. Özer, On Real Quadratic Number Fields Related With Specific Type of Continued Fractions, J. Anal. Number Theory, 4 (2016), 85-90. doi: 10.18576/jant/040201
[16]	Ö. Özer, Notes On Especial Continued Fraction Expansions and Real Quadratic Number Fields, J. Eng. Sci., 2 (2016), 74-89.
[17]	Ö. Özer, Fibonacci Sequence and Continued Fraction Expansions in Real Quadratic Number Fields, Malays. J. Math. Sci., 11 (2017), 97-118.
[18]	Ö. Özer, On The Fundamental Units of Certain Real Quadratic Number Fields, Karaelmas Sci. Eng. J., 7 (2017), 160-164.
[19]	Ö. Özer, A. B. M. Salem, A Computational Technique For Determining Fundamental Unit in Explicit Type of Real Quadratic Number Fields, Int. J. Adv. Appl. Sci., 4 (2017), 22-27. doi: 10.21833/ijaas.2017.02.004
[20]	Ö. Özer, A Study On The Fundamental Unit of Certain Real Quadratic Number Fields, Turk. J. Anal. Number Theory (TJANT), 6 (2018), 1-8. doi: 10.12691/tjant-6-1-1
[21]	Ö. Özer, B. Djamel, Some Results on Special Types of Real Quadratic Fields, Bulletin of the Karaganda Univeristy-Mathematics, 3 (2019), 48-58. doi: 10.31489/2019M1/48-58
[22]	Ö. Özer, A Handy Technique For Fundamental Unit in Specific Type of Real Quadratic Fields, Appl. Math. Nonlinear Sci. (AMNS), 4 (2019), 1-4. doi: 10.2478/AMNS.2019.1.00001
[23]	O. Perron, Die Lehre von den Kettenbrchen, Band I: Elementare Kettenbrche, 1977.
[24]	R. Sasaki, A characterization of certain real quadratic fields, Proc. Japan Acad. Series A, 62 (1986), 97-100. doi: 10.3792/pjaa.62.97
[25]	W. Sierpinski, Elementary Theory of Numbers, Warsaw, Monogra Matematyczne, 1964.
[26]	K. Tomita, Explicit representation of fundamental units of some quadratic fields, Proc. Japan Acad. Series A, 71 (1995), 41-43. doi: 10.3792/pjaa.71.41
[27]	K. Tomita, K. Yamamuro, Lower bounds for fundamental units of real quadratic Fields, Nagoya Math. J., 166 (2002), 29-37. doi: 10.1017/S0027763000008230
[28]	K. S. Williams, N. Buck, Comparison of the lengths of the continued fractions of $\sqrt{d}$ and $\frac{1+\sqrt{d}}{2}$, Proc. Amer. Math. Soc., 120 (1994), 995-1002.
[29]	H. Yokoi, The fundamental unit and class number one problem of real quadratic fields with prime discriminant, Nagoya Math. J., 120 (1990), 51-59. doi: 10.1017/S0027763000003238
[30]	H. Yokoi, A note on class number one problem for real quadratic fields, Proc. Japan Acad. Series A, 69 (1993), 22-26. doi: 10.3792/pjaa.69.22
[31]	H. Yokoi, The fundamental unit and bounds for class numbers of real quadratic fields, Nagoya Math. J., 124 (1991), 181-197. doi: 10.1017/S0027763000003834
[32]	H. Yokoi, New invariants and class number problem in real quadratic fields, Nagoya Math., J., 132 (1993), 175-197. doi: 10.1017/S0027763000004700
[33]	Z. Zhang, Q. Yue, Fundamental units of real quadratic fields of odd class number, J. Number Theory, 137 (2014), 122-129. doi: 10.1016/j.jnt.2013.10.019

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