Geodesic
Chebyshev polynomials and Pell equations over finite fields
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 491-510.

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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 $x^2-my^2=n$.
DOI : 10.21136/CMJ.2020.0451-19
Classification : 11D09, 11D79, 11T99, 12E10, 12E20
Keywords: finite field; Chebyshev polynomial; Pell equation 
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     title = {Chebyshev polynomials and {Pell} equations over finite fields},
     journal = {Czechoslovak Mathematical Journal},
     pages = {491--510},
     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0451-19/}
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Cohen, Boaz. Chebyshev polynomials and Pell equations over finite fields. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 491-510. doi : 10.21136/CMJ.2020.0451-19. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0451-19/

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