Identification of Patterns of Continued Fractions of √s – [√s] Where is a Square Free Positive Number

A. Gnanam; S. Krithika

doi:https://doi.org/10.14445/22315373/IJMTT-V54P532

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 54 | Number 4 | Year 2018 | Article Id. IJMTT-V54P532 | DOI : https://doi.org/10.14445/22315373/IJMTT-V54P532

Identification of Patterns of Continued Fractions of √s – [√s] Where is a Square Free Positive Number

A. Gnanam, S. Krithika

Abstract

It is observed in [3] any periodic continued fraction represents a quadratic irrational and vice versa. In this paper we try to identify the patterns of continued fractions of √s – [√s] where S is a square free positive number.

Keywords

Continued fractions, Simple continued fraction, Periodic continued fraction, Quadratic irrational, Euclidean algorithm, Bisection method, False Position method, Iteration method and Newton -Raphson method.

References

[1] A. Ya. Khinchin, “Continued Fractions” , Dovers books on mathematics,1997.
[2] George E. Andrews, “Number Theory”, W.B. Saunders Company.
[3] Ivan Niven , Herbert S. Zuckerman, Hugh L. Montgomery, “An introduction to theory of numbers”, Fifth edition, Wiley Student Edition.
[4] Jonathan Browein, Alfvander Poorten, Jeffrey Shallit, Wadim Zudilin, “Neverending Fractions An Introduction to continued fractions”, Cambridge University Press.
[5] Neville Robbins, “Beginning Number Theory”, Second edition, Naros, Publishing House.
[6] Olds, C.D., “Continued Fractions”, Random House: New York, 1963.
[7] S.S.Sastry, Introductory methods of Numerical Analysis , Fifth edition, PHI Private ltd. New Delhi, 2012.
[8] T.K. Manicavachagom Pillai, T. Natarajan, K. S. Ganapathy “Algebra” Vol.II, S. Viswanathan Pvt., Ltd., Chennai
[9] Continued fractions on web: http://archives.math.utk.edu/ atuyl/confrac/history.html.
[10] Continued fractions on web: http://archives.maths.survey.ac.uk./hosted sites/R.Knott/Fibonacci/cfINTRO.htm#section 1.

Citation :

A. Gnanam, S. Krithika, "Identification of Patterns of Continued Fractions of √s – [√s] Where is a Square Free Positive Number," International Journal of Mathematics Trends and Technology (IJMTT), vol. 54, no. 4, pp. 281-288, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V54P532