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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, 𝒰(R) be the group of units of R and k be a positive integer. We say that a 𝒰(R) is a k-unit if ak = 1. In particular, if the ring  R is n for some positive integer n we say that a is a k-unit modulo n. We denote by 𝒰k(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.

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