Comptes Rendus
Number Theory
Empirical estimates of the average orders of orbits period lengths in Euler groups
Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 15-20.

The averaged growth rate of period's length of the geometrical progressions {q t mod n,t=0,1,...} for increasing n is empirically estimated for different values of q. The experimental results, obtained for n up to 106, allow us to conjecture that the average order of period's length is Cn ln (n), where constant C depends on q.

On donne une estimation expérimentale du taux moyen de croissance de la longueur de la période des progressions géométriques {q t mod n,t=0,1,...} pour n croissant, pour des valeurs différentes de q. Les résultats empiriques, obtenus pour n jusqu'à 106, permettent de conjecturer que l'ordre moyen de la longueur de la période est Cn ln (n), où la constante C dépend de q.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.02.021
Francesca Aicardi 1

1 Sistiana Mare 56 pr, 34019 Trieste, Italy
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     title = {Empirical estimates of the average orders of orbits period lengths in {Euler} groups},
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Francesca Aicardi. Empirical estimates of the average orders of orbits period lengths in Euler groups. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 15-20. doi : 10.1016/j.crma.2004.02.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.02.021/

[1] V.I. Arnold Euler Groups and Arithmetics of Geometric Progressions, MCMME, Moscow, 2003 (40 p)

[2] V.I. Arnold Fermat–Euler dynamical system and statistics of the geometric progressions, Funct. Anal. Appl., Volume 37 (2002) no. 1, pp. 1-20

[3] V.I. Arnold Ergodic arithmetical properties of the dynamics of geometric progressions, Moscow Math. J. (2003)

[4] V.I. Arnold Russian Math. Surveys, 58 (2003) no. 4

[5] V.I. Arnold Weak asymptotics of the solutions numbers of Diophantine problems, Funct. Anal. Appl., Volume 37 (1999) no. 3, pp. 65-66

[6] P.G. Dirichlet, Abhand. Ak. Wiss., Berlin (Math.), 1849, pp. 78–81; Werke, II, pp. 60–64

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