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

### John H. Castillo and Jhony Fernando Caranguay Mainguez

Vol. 15 (2022), No. 3, 367–378
##### Abstract

Let $R$ be a ring with identity, $\mathsc{𝒰}\left(R\right)$ be the group of units of $R$ and $k$ be a positive integer. We say that $a\in \mathsc{𝒰}\left(R\right)$ is a $k$-unit if ${a}^{k}=1$. In particular, if the ring  $R$ is ${ℤ}_{n}$ for some positive integer $n$ we say that $a$ is a $k$-unit modulo $n$. We denote by ${\mathsc{𝒰}}_{k}\left(n\right)$ the set of $k$-units modulo $n$. We represent the number of $k$-units modulo $n$ by ${\mathrm{du}}_{k}\left(n\right)$ and the ratio of $k$-units modulo $n$ by ${\mathrm{rdu}}_{k}\left(n\right)=\varphi \left(n\right)∕{\mathrm{du}}_{k}\left(n\right)$, where $\varphi$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation ${\mathrm{rdu}}_{2}\left(n\right)=1$ are the divisors of $24$. Our main result finds all positive integers $n$ such that ${\mathrm{rdu}}_{k}\left(n\right)=1$ for a given $k$. Then we connect this equation with the Carmichael numbers and two of their generalizations, namely, Knödel numbers and generalized Carmichael numbers.

##### Keywords
diagonal property, diagonal unit, unit set of a ring, $k$-unit, Carmichael number, Knödel number, Carmichael generalized number
##### Mathematical Subject Classification 2010
Primary: 11A05, 11A07, 11A15, 16U60