# Extreme Covering Systems of the Integers

## TL;DR

This paper establishes lower bounds on the largest modulus and the least common multiple for covering systems of integers with specific minimal moduli, providing optimal constants for these bounds.

## Contribution

It proves new lower bounds on the largest modulus and LCM in covering systems with minimal moduli 3 and 4, establishing their optimality.

## Key findings

- Largest modulus at least 60 when minimal modulus is 4
- LCM at least 120 when minimal modulus is 3
- LCM at least 360 when minimal modulus is 4

## Abstract

It is proved 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 LCM of the moduli is at least 120; finally, if the least modulus is 4, the LCM of the moduli is at least 360. The constants 60, 120, and 360 can not be replaced by larger constants.