#### Date of Award

Spring 2022

#### Document Type

Open Access Thesis

#### Department

Mathematics

#### First Advisor

Ognian Trifonov

#### Second Advisor

Michael Filaseta

#### Abstract

A covering system or a covering is a set of linear congruences such that every integer satisfies at least one of these congruences. In 1950, Erdős posed a problem regarding the existence of a finite covering with distinct moduli and an arbitrarily large minimum modulus. This remained unanswered until 2015 when Robert Hough proved an explicit bound of 1016 for the minimum modulus of any such covering. In this thesis, we examine the use of covering systems in number theory results, expand upon the proof of the existence of an upper bound on the minimum modulus in the case of distinct square-free moduli, and give a sharper bound of 118 for the minimum modulus of a finite covering with distinct square-free moduli.

#### Recommended Citation

Cummings, M. C.(2022). *Covering Systems and the Minimum Modulus Problem.* (Master's thesis). Retrieved from https://scholarcommons.sc.edu/etd/6665