# On covering systems of integers

## TL;DR

This paper investigates the structure of covering systems of integers, establishing that such systems must include a modulus divisible by a prime ≤ 19, extending previous results and addressing open questions about odd moduli.

## Contribution

It generalizes Hough's negative result on minimum moduli by showing all covering systems have a modulus divisible by a prime ≤ 19.

## Key findings

- Every covering system has a modulus divisible by a prime ≤ 19
- Previous results showed divisibility by 2 or 3, now extended to 19
- Addresses open problem about covering systems with all odd moduli

## Abstract

A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of residues modulo elements of $S$ covering the integers. Hough has shown that if the smallest modulus in $S$ is at least $10^{16}$, then there is none. However, the question of whether there is a covering of the integers with all odd moduli remains open. We consider multiplicative restrictions on the set of moduli to generalize Hough's negative solution to the minimum modulus problem. In particular, we find that every covering system of the integers has a modulus divisible by a prime number less than or equal to $19$. Hough and Nielsen have shown that every covering system has a modulus divisible by either $2$ or $3$.