Geodesic
Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 1-15.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FAA_2004_38_1_a0,
     author = {V. I. Arnol'd},
     title = {Fermat {Dynamics,} {Matrix} {Arithmetics,} {Finite} {Circles,} and {Finite} {Lobachevsky} {Planes}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--15},
     publisher = {mathdoc},
     volume = {38},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a0/}
}
TY  - JOUR
AU  - V. I. Arnol'd
TI  - Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2004
SP  - 1
EP  - 15
VL  - 38
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a0/
LA  - ru
ID  - FAA_2004_38_1_a0
ER  - 
%0 Journal Article
%A V. I. Arnol'd
%T Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2004
%P 1-15
%V 38
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a0/
%G ru
%F FAA_2004_38_1_a0
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/

[1] Arnold V., Avez A., Problèmes Ergodiques de la Mécanique Classique, Gautier–Villars, Paris, 1967 | MR

[2] Arnold V. I., “Topologiya i statistika formul arifmetiki”, UMN, 58:4 (352) (2003), 3–28 | DOI | MR

[3] Arnold V. I., “Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world”, Bull. Braz. Math. Soc., N.S., 34:1 (2003), 1–41 ; (2002), MCCME, Dubna | DOI | MR

[4] Girard A., Invention nouvelle en l'algèbre, Amsterdam, 1629

[5] Newton I., Arithmetica universalis, Cambridge, 1707, 57–63