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2021 The Erdős–Selfridge problem with square-free moduli
Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, Marius Tiba
Algebra Number Theory 15(3): 609-626 (2021). DOI: 10.2140/ant.2021.15.609

Abstract

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all odd. We show that if in addition one assumes the moduli are square-free, then there must be an even modulus.

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Paul Balister. Béla Bollobás. Robert Morris. Julian Sahasrabudhe. Marius Tiba. "The Erdős–Selfridge problem with square-free moduli." Algebra Number Theory 15 (3) 609 - 626, 2021. https://doi.org/10.2140/ant.2021.15.609

Information

Received: 31 January 2019; Revised: 14 August 2020; Accepted: 18 September 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/ant.2021.15.609

Subjects:
Primary: 11B25
Secondary: 11A07 , 11N35

Keywords: covering systems , Erdős–Selfridge problem

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 3 • 2021
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