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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

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.

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
Received: 5 January 2020
Revised: 28 October 2021
Accepted: 31 October 2021
Published: 2 December 2022

Communicated by Kenneth S. Berenhaut
John H. Castillo
Departamento de Matemáticas y Estadística
Universidad de Nariño
San Juan de Pasto
Jhony Fernando Caranguay Mainguez
Departamento de Matemáticas y Estadística
Universidad de Nariño
San Juan de Pasto