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

TL;DR

This paper investigates the properties of the multiplicative and projective multiplicative order functions of $a^n$ modulo $n$, establishing relationships that reduce their computation to the square-free case, with applications in Steinhaus triangles.

Contribution

It proves relationships between these functions for integers with the same square-free part, simplifying their analysis to the square-free case, and connects to problems in Steinhaus triangles.

Findings

Established relationships between $\alpha_{n_1}$ and $\alpha_{n_2}$ for same square-free part.

Reduced the problem of determining $\alpha_n$ and $\beta_n$ to the square-free case.

Connected the functions to the study of balanced Steinhaus triangles in $\\mathbb{Z}/n\\mathbb{Z}$.

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.