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

Jonathan Chappelon (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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.

Mots clés: multiplicative order; projective multiplicative order; balanced Steinhaus triangles; Steinhaus triangles; Molluzzo's problem

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]

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