Geodesic
Fermat--Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 1, pp. 1-18

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Let $n$ be an integer. A Fermat–Euler dynamical system acts on the set of mod-$n$ residues coprime to $n$ by multiplication by a constant (which is also coprime to $n$). We study the dependence of the period and the number of orbits of this dynamical system on $n$. Theorems generalizing Fermat's little theorem, as well as empirical conjectures, are given.
Keywords: Euler function, Fermat's little theorem, chaotic behavior, weak asymptotics, quadratic residue, geometric progression, Young diagram. 
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     author = {V. I. Arnol'd},
     title = {Fermat--Euler {Dynamical} {Systems} and the {Statistics} of {Arithmetics} of {Geometric} {Progressions}},
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V. I. Arnol'd. Fermat--Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 1, pp. 1-18. http://geodesic.mathdoc.fr/item/FAA_2003_37_1_a0/