Geodesic
Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 1-15 Cet article a éte moissonné depuis la source Math-Net.Ru

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Congruences generalizing Fermat's little theorem are proved for the traces of powers of integer matrices. Their relations to Lobachevsky geometries over finite fields and combinatorics of the matrix squaring operation as well as to the corresponding Riemann surfaces with their Kepler cubes are discussed.
Keywords: arithmetics, symmetric function, de Sitter world, trace, Fermat's little theorem, Lobachevsky geometry, Kepler cube, Riemann surface. 
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V. I. Arnol'd. Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 1-15. http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a0/

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