A further investigation on covering systems with odd moduli

TL;DR

This paper explores a variant of the odd covering problem, allowing one odd integer to appear multiple times as a modulus while others are distinct, advancing understanding of covering systems with odd moduli.

Contribution

It introduces a new variant of the odd covering problem permitting repeated moduli, expanding the scope of covering system investigations.

Findings

Analyzed the implications of allowing repeated odd moduli in covering systems.

Provided new insights into the structure of covering systems with odd moduli.

Extended previous results by considering non-distinct moduli in the system.

Abstract

Erd\H{o}s first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.