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. 
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/
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