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

Chappelon, Jonathan

Journal of integer sequences, Tome 13 (2010) no. 2.

Résumé

Summary: 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 a 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 $Z/n$Z for $n$ odd.

Détail

Keywords: multiplicative order, projective multiplicative order, balanced Steinhaus triangles, Steinhaus triangles, Molluzzo's problem