Czechoslovak Mathematical Journal, first online, pp. 1-20


Chebyshev polynomials and Pell equations over finite fields

Boaz Cohen

Received October 10, 2019.   Published online December 9, 2020.

Abstract:  We shall describe how to construct a fundamental solution for the Pell equation $x^2-my^2=1$ over finite fields of characteristic $p\neq2$. Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^2-my^2=n$.
Keywords:  finite field; Chebyshev polynomial; Pell equation
Classification MSC:  12E20, 11D09, 12E10, 11D79, 11T99
DOI:  10.21136/CMJ.2020.0451-19

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] A. T. Benjamin, D. Walton: Counting on Chebyshev polynomials. Math. Mag. 82 (2009), 117-126. DOI 10.1080/0025570X.2009.11953605 | MR 2512595 | Zbl 1223.33013
[2] K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84. Springer, New York (1990). DOI 10.1007/978-1-4757-2103-4 | MR 1070716 | Zbl 0712.11001
[3] W. J. LeVeque: Topics in Number Theory. Vol I. Dover Publications, Mineola (2002). MR 1942365 | Zbl 1009.11001

Affiliations:   Boaz Cohen, Department of Computer Science, The Academic College of Tel-Aviv, Rabenu Yeruham St., P.O.Box 8401 Tel-Aviv Yaffo, 6818211, Israel, e-mail: arctanx@gmail.com


 
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