Computations and observations on congruence covering systems

TL;DR

This paper classifies all distinct covering systems with up to ten moduli and determines that the smallest such system with all moduli greater than 2 has size 11, advancing understanding of covering systems.

Contribution

It provides a complete classification of small distinct covering systems and establishes the minimal size for systems with moduli exceeding 2.

Findings

Complete classification of distinct covering systems with ≤10 moduli

Minimum size of such systems with moduli > 2 is 11

Groupings based on equivalence classes

Abstract

A $\textit{covering system}$ is a collection of integer congruences such that every integer satisfies at least one congruence in the collection. A covering system is called $\textit{distinct}$ if all of its moduli are distinct. An expansive literature has developed on covering systems since their introduction by Erd\H{o}s. Here we provide a full classification of distinct covering systems with at most ten moduli, which we group together based on two forms of equivalence. As a consequence, we determine the minimum cardinality of a distinct covering system with all moduli exceeding $2$, which is $11$.