Covering systems where the prime divisors of all moduli are only $2$, $3$, or $5$

TL;DR

This paper classifies specific covering systems with moduli whose prime factors are only 2, 3, or 5, providing complete descriptions for certain cases and constructing examples for others.

Contribution

It offers a complete characterization of quadruples defining such covering systems for small minimum moduli and introduces methods to establish nonexistence.

Findings

Complete description for m=2,3,4,5, or 6 (except one case)

Constructed a covering system with m=8, a=8, b=3, c=2

Established nonexistence using integer programming and density estimates

Abstract

We try to find all quadruples of positive integers $(m,a,b,c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexistence of covering systems is established via integer programming or by using a new estimate on the density of a set covered by a system of congruences.