On Primitive Covering Numbers

TL;DR

This paper investigates primitive covering numbers, providing a formula for counting certain coverings and disproving a conjecture by Sun by constructing counterexamples.

Contribution

It derives an exact counting formula for coverings with a given primitive covering number and disproves Sun's conjecture by constructing counterexamples.

Findings

Derived a formula for the number of coverings with a given primitive covering number.

Constructed an infinite set of primitive covering numbers that do not meet Sun's conjecture.

Disproved Sun's conjecture regarding primitive covering numbers.

Abstract

In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called \emph{primitive} if no proper divisor of $L$ is a covering number. Sun constructed an infinite set $\mathcal L$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\in \mathcal L$, we derive a formula that gives the exact number of coverings that have $L$ as the least common multiple of the set $M$ of moduli, under certain restrictions on $M$. Additionally, we disprove Sun's conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.