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Keywords:
finite field; Chebyshev polynomial; Pell equation
finite field; Chebyshev polynomial; Pell equation
Summary:
We shall describe how to construct a fundamental solution for the Pell equation $x^2-my^2=1$ over finite fields of characteristic $p\neq 2$. 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 $x^2-my^2=n$.
References:
[1] Benjamin, A. T., Walton, D.: Counting on Chebyshev polynomials. Math. Mag. 82 (2009), 117-126. DOI 10.1080/0025570X.2009.11953605 | MR 2512595 | Zbl 1223.33013
[2] Ireland, K., Rosen, M.: 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] LeVeque, W. J.: Topics in Number Theory. Vol I. Dover Publications, Mineola (2002). MR 1942365 | Zbl 1009.11001
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