Exact coverings of the integers by arithmetic progressions
type: Item
instance of: scholarly article
title: Exact coverings of the integers by arithmetic progressions
zbMATH Open document ID: 0591.10047
DOI: 10.1016/0012-365X(86)90079-8
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A system of arithmetic progressions is said to be exactly covering (ECS) if each integer belongs to exactly one progression of that system. An ECS in which every modulus (difference) occurs at most M times is denoted by ECS(M). Main theorem: Let \(p_ 1<...<p_ t\) be the distinct prime divisors of the moduli in an ECS(M), then \(p_ t\leq M\prod^{t-1}_{i=1}p_ i/(p_ i-1).\) This is a generalization of a theorem by N. Burshtein, who proved similar results for a special case of so called natural exactly covering systems. Corollary (4.1). The greatest prime \(p_ t\) dividing the modulus of any arithmetic progression in an ECS(M) is at most equal to the greatest prime \(p^*\) satisfying \(M\prod_{p<p^*}p/(p-1)\geq p^*.\) This represents for the case \(M=2\) the proof of a conjecture by Burshtein: all moduli of an ECS(2) are of the form \(2^ a 3^ b 5^ c 7^ d.\)
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 3948406
zbMATH Keywords: upper and lower bounds on prime divisors of moduli
zbMATH Keywords: upper bound on
zbMATH Keywords: least modulus
zbMATH Keywords: system of arithmetic progressions
zbMATH Keywords: exactly covering systems
author: Jamie Simpson
reviewed by: Štefan Znám
MaRDI profile type: Publication
cites work: On natural exactly covering systems of congruences having moduli occurring at most \(M\) times
cites work: On natural exactly covering systems of congruences having moduli occurring at most twice
cites work: Q3909098
cites work: Q4145769
full work available at URL: 0012-365x(86)90079-8
OpenAlex ID: W2133031537
description: scientific article; zbMATH DE number 3948406