On the multiplicative order of $a^n$ modulo $n$

Abstract : Let $n$ be a positive integer and $\alpha_n$ be the arithmetic function which assigns the multiplicative order of $a^n$ modulo $n$ to every integer $a$ coprime to $n$ and vanishes elsewhere. Similarly, let $\beta_n$ assign the projective multiplicative order of $a^n$ modulo $n$ to every integer $a$ coprime to $n$ and vanishes elsewhere. In this paper, we present a study of these two arithmetic functions. In particular, we prove that for positive integers $n_1$ and $n_2$ with the same square-free part, there exists an exact relationship between the functions $\alpha_{n_1}$ and $\alpha_{n_2}$ and between the functions $\beta_{n_1}$ and $\beta_{n_2}$. This allows us to reduce the determination of $\alpha_n$ and $\beta_n$ to the case where $n$ is square-free. These arithmetic functions recently appeared in the context of an old problem of Molluzzo, and more precisely in the study of which arithmetic progressions yield a balanced Steinhaus triangle in $\mathbb{Z}/n\mathbb{Z}$ for $n$ odd.
Keywords :
Type de document :
Article dans une revue
Journal of Integer Sequences, University of Waterloo, 2010, 13, pp.Article 10.2.1
Domaine :
Liste complète des métadonnées

Littérature citée [3 références]

https://hal.archives-ouvertes.fr/hal-00371233
Contributeur : Jonathan Chappelon <>
Soumis le : mardi 22 mars 2016 - 16:23:04
Dernière modification le : mardi 26 avril 2016 - 17:56:47
Document(s) archivé(s) le : jeudi 23 juin 2016 - 16:19:59

Fichiers

Multiplicative_order_FinalVers...
Fichiers produits par l'(les) auteur(s)

Identifiants

• HAL Id : hal-00371233, version 1
• ARXIV : 0902.4366

Citation

Jonathan Chappelon. On the multiplicative order of $a^n$ modulo $n$. Journal of Integer Sequences, University of Waterloo, 2010, 13, pp.Article 10.2.1. 〈hal-00371233〉

Métriques

Consultations de la notice

118

Téléchargements de fichiers