A stronger connection between the Erd\H{o}s-Burgess and Davenport constants

TL;DR

This paper establishes a new link between the Erd ext{"o}s-Burgess and Davenport constants for certain algebraic structures, confirming a conjecture for rings with up to two prime factors and exploring extensions to other rings.

Contribution

It confirms a conjecture relating Erd ext{"o}s-Burgess and Davenport constants for rings with at most two prime factors, and extends techniques to other rings.

Findings

Confirmed the conjecture for rings with up to two prime factors.

Established a stronger connection between the two constants in specific algebraic settings.

Discussed potential extensions of the methods to broader classes of rings.

Abstract

The Erd\H{o}s-Burgess constant of a semigroup $S$ is the smallest positive integer $k$ such that any sequence over $S$ of length $k$ contains a nonempty subsequence whose elements multiply to an idempotent element of $S$. In the case where $S$ is the multiplicative semigroup of $\mathbb{Z}/n\mathbb{Z}$, we confirm a conjecture connecting the Erd\H{o}s-Burgess constant of $S$ and the Davenport constant of $(\mathbb{Z}/n\mathbb{Z})^{\times}$ for $n$ with at most two prime factors. We also discuss the extension of our techniques to other rings.