The set of $k$-units modulo $n$

TL;DR

This paper investigates the structure and properties of $k$-units modulo $n$, characterizing integers where the ratio of $k$-units to the Euler totient is 1, and explores connections to Carmichael and related numbers.

Contribution

It characterizes all positive integers $n$ for which the ratio of $k$-units to Euler's totient is 1, extending previous results and linking to Carmichael-type numbers.

Findings

Identifies all $n$ with $ ext{rdu}_k(n)=1$ for given $k$.

Establishes connections between $k$-units and Carmichael, Kn"odel, and generalized Carmichael numbers.

Provides formulas and characterizations for $k$-units modulo $n$.

Abstract

Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{\phi(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $\phi$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation $\text{rdu}_2(n)=1$ are the divisors of $24$. The main result of this work, is that for a given $k$, we find the positive integers $n$ such that $\text{rdu}_k(n)=1$. Finally, we give some connections of this equation with Carmichael's numbers and two of its generalizations: Kn\"odel numbers and generalized Carmichael numbers.