Irreducibles in the Integers modulo n

TL;DR

This paper investigates the structure of irreducible elements in the integers under modular equivalence, providing specific results for certain n and aiming for a general characterization for prime n.

Contribution

It identifies irreducible integers modulo n for specific cases and advances toward a general description for all prime n.

Findings

Irreducible integers modulo specific n identified

Generalization towards prime n cases initiated

Foundations laid for a comprehensive characterization

Abstract

For an element $a$ of an integral domain D under an equivalence relation \tau, the \tau-factorization of a is defined as \lambda a_1 a_2... a_k, where \lambda is a unit in D and a_i \tau a_j for all i, j. An irreducible element has no proper \tau-factorization; that is, a \tau-factorization in which there is more than one distinct non-unit factor. In this paper, the irreducible integers under the congruence modulo n relation for some values of n are found, and these findings are generalized in the first step toward a general characterization of the irreducible integers under this relation for any prime n.