ON SOME PROPERTIES OF THE MOD $n$ SEQUENCES OF $n$-TH POWERS

Abstract

We express the prime factorization of a composite number $n$ as $n=p_1^{e_1} \cdots p_r^{e_r}$ with distinct prime numbers $p_1, \ldots, p_r$ and positive integers $e_1, \ldots, e_r$. In this article, we prove that the sequence $\left\{a^n(\bmod n)\right\}_{a=0}^{n-1}$ consists of $\left(p_1^{e_1-1} \cdots p_r^{e_r-1}\right)$-copies of $\left\{a^n(\bmod n)\right\}_{a=0}^{p_1 \cdots p_r-1}$. This result is a generalization of Theorem 6.1 (resp. Theorem 6.2) in [1] for the case where $n$ is of the form $p^k$ with a prime number $p$ and a positive integer $k$ (resp. $p^k q$ with a prime number $q \neq p$ ).

Received: June 12, 2024
Revised: July 25, 2024
Accepted: August 8, 2024