On the periodicity of irreducible elements in arithmetical congruence monoids

TL;DR

This paper investigates the pattern and periodicity of irreducible elements in arithmetical congruence monoids, revealing conditions under which their distribution becomes eventually periodic based on parameters a and b.

Contribution

It characterizes when the set of irreducible elements in these monoids exhibits eventual periodicity depending on the parameters a and b.

Findings

Irreducible elements can form an eventually periodic sequence.

The periodicity depends on specific relations between a and b.

Conditions for periodicity are explicitly characterized.

Abstract

Arithmetical congruence monoids, which arise in non-unique factorization theory, are multiplicative monoids $M_{a,b}$ consisting of all positive integers $n$ satsfying $n \equiv a \bmod b$. In this paper, we examine the asymptotic behavior of the set of irreducible elements of $M_{a,b}$, and characterize in terms of $a$ and $b$ when this set forms an eventually periodic sequence.