Comptes Rendus
no. 1
Number Theory
Empirical estimates of the average orders of orbits period lengths in Euler groups
Vladimir Arnold
Francesca Aicardi1
1Sistiana Mare 56 pr, 34019 Trieste, Italy
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 
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
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