2014/06/26 by Lenny Jones, Daniel White, Jones, Lenny +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1406.6851
openalex publication_date 2014/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2007, Zhi-Wei Sun defined a 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 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∈ \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.