The set of k-units modulo n

Castillo, John H.; Caranguay Mainguez, Jhony Fernando

doi:10.2140/involve.2022.15.367

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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

DOI: 10.2140/involve.2022.15.367

Abstract

Let $R$ be a ring with identity, $𝒰 (R)$ be the group of units of $R$ and $k$ be a positive integer. We say that $a \in 𝒰 (R)$ 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 $𝒰_{k} (n)$ the set of $k$ -units modulo $n$ . We represent the number of $k$ -units modulo $n$ by ${du}_{k} (n)$ and the ratio of $k$ -units modulo $n$ by ${rdu}_{k} (n) = ϕ (n) ∕ {du}_{k} (n)$ , where $ϕ$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation ${rdu}_{2} (n) = 1$ are the divisors of $24$ . Our main result finds all positive integers $n$ such that ${rdu}_{k} (n) = 1$ 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.

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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

Milestones

Received: 5 January 2020

Revised: 28 October 2021

Accepted: 31 October 2021

Published: 2 December 2022

Communicated by Kenneth S. Berenhaut

Authors

John H. Castillo
Departamento de Matemáticas y Estadística Universidad de Nariño San Juan de Pasto Colombia

Jhony Fernando Caranguay Mainguez
Departamento de Matemáticas y Estadística Universidad de Nariño San Juan de Pasto Colombia

Vol. 15, No. 3, 2022