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

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

2009-02-25

arxiv.org/abs/0902.4366v2

Mathematics

Number Theory

14 pages, 2 figures

Scientific paper

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 vanish elsewhere. In this paper, we present a study of these two arithmetic functions. In particular, we prove that for every 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.

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