#### Date of Award

Spring 2023

#### Document Type

Open Access Dissertation

#### Department

Mathematics

#### First Advisor

Ognian Trifonov

#### Abstract

This dissertation considers three different topics.

In the first part, we prove that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also, if the least modulus is 3, the least common multiple of the moduli is at least 120; finally, if the least modulus is 4, the least common multiple of the moduli is at least 360. The constants 60, 120, and 360 are best possible, they cannot be replaced by larger constants. We also show that there do not exist distinct covering systems with all of the moduli in the interval [n, 9n] for n ≥ 3.

In the second part, we obtain a lower bound for the maximum distance between any three distinct points in an affine lattice which are close to a helix with small curvature and torsion. This is a generalization of analogous results in two dimensions.

In the final part, we prove that every positive integer n which is not equal to 1, 2, 3, 6, 11, 30, 155, or 247 can be represented as a sum of a squarefree number and a prime not exceeding √ n.

#### Recommended Citation

Dalton, J. R.(2023). *Extreme Covering Systems, Primes Plus Squarefrees, and Lattice Points Close to a Helix.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/7255