NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS

Özen Özer

NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS

Abstract

The primary purpose of this paper is to classify real quadratic fields Q(√d) which

include the form of specific continued fraction expansion of integral basis element 𝑤𝑑 for

arbitrary period length ℓ = ℓ(𝑑) where d ≡ 2,3(mod4) is a square free positive integers.

Furthermore, the present paper deals with determining new certain parametric

formulas of fundamental unit............ and Yokoi’s d-invariants nd , md for

such real quadratic fields. All results are also supported by several numerical tabular forms.

Keywords

Quadratic Fields,Continued Fractions,Fundamental Units.

References

[1] R.A. Mollin, “Quadratics”, Boca Rato, F.L, CRC Press, 1996.
[2] C.D. Olds, “Continued Functions” ,New York: Random House, 1963.
[3] O. Perron, “Die Lehre von den Kettenbrüchen” ,New York: Chelsea, Reprint from Teubner Leipzig, 1950.
[4] R. Sasaki, “A characterization of certain real quadratic fields”, Proc. Japan Acad, 62, Ser. A, 1986, no. 3, 97-100.
[5] W. Sierpinski, “Elementary Theory of Numbers”, Warsaw: Monografi Matematyczne, 1964.
[6] K.Tomita, “Explicit representation of fundamental units of some quadratic fields”, Proc. Japan Acad., 71, Ser. A, 1995, no. 2, 41-43.
[7] K. Tomita and K. Yamamuro, “Lower bounds for fundamental units of real quadratic fields”, Nagoya Math. J, Vol.166, 2002, 29-37.
[8] K.S. Williams and N. Buck, “Comparison of the lengths of the continued fractions of √D and 1/2(1 + √D)”, Proc. Amer. Math. Soc., 120 no. 4, 1994, 995-1002.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Özen Özer ^* This is me

Publication Date

June 30, 2016

Submission Date

February 19, 2016

Acceptance Date

-

Published in Issue

Year 2016 Volume: 2 Number: 1

IZ

https://izlik.org/JA82XU44BZ

Cite

RIS / Bibtex

APA

Özer, Ö. (2016). NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS. Kirklareli University Journal of Engineering and Science, 2(1), 74-89. https://izlik.org/JA82XU44BZ

AMA

1.Özer Ö. NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS. Kirklareli University Journal of Engineering and Science. 2016;2(1):74-89. https://izlik.org/JA82XU44BZ

Chicago

Özer, Özen. 2016. “NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS”. Kirklareli University Journal of Engineering and Science 2 (1): 74-89. https://izlik.org/JA82XU44BZ.

EndNote

Özer Ö (June 1, 2016) NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS. Kirklareli University Journal of Engineering and Science 2 1 74–89.

IEEE

[1]Ö. Özer, “NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS”, Kirklareli University Journal of Engineering and Science, vol. 2, no. 1, pp. 74–89, June 2016, [Online]. Available: https://izlik.org/JA82XU44BZ

ISNAD

Özer, Özen. “NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS”. Kirklareli University Journal of Engineering and Science 2/1 (June 1, 2016): 74-89. https://izlik.org/JA82XU44BZ.

JAMA

1.Özer Ö. NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS. Kirklareli University Journal of Engineering and Science. 2016;2:74–89.

MLA

Özer, Özen. “NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS”. Kirklareli University Journal of Engineering and Science, vol. 2, no. 1, June 2016, pp. 74-89, https://izlik.org/JA82XU44BZ.

Vancouver

1.Özen Özer. NOTES ON ESPECIAL CONTINUED FRACTION EXPANSIONS AND REAL QUADRATIC NUMBER FIELDS. Kirklareli University Journal of Engineering and Science [Internet]. 2016 Jun. 1;2(1):74-89. Available from: https://izlik.org/JA82XU44BZ