The Erdős–Selfridge problem with square-free moduli

Balister, Paul; Bollobás, Béla; Morris, Robert; Sahasrabudhe, Julian; Tiba, Marius

doi:10.2140/ant.2021.15.609

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The Erdős–Selfridge problem with square-free moduli

Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba

Vol. 15 (2021), No. 3, 609–626

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.

Keywords

covering systems, Erdős–Selfridge problem

Mathematical Subject Classification 2010

Primary: 11B25

Secondary: 11A07, 11N35

Milestones

Received: 31 January 2019

Revised: 14 August 2020

Accepted: 18 September 2020

Published: 20 May 2021

Authors

Paul Balister
University of Oxford Oxford United Kingdom

Béla Bollobás
University of Cambridge Cambridge United Kingdom	University of Memphis Memphis, TN United States

Robert Morris
IMPA Rio de Janeiro Brazil

Julian Sahasrabudhe
University of Cambridge Cambridge United Kingdom

Marius Tiba
University of Cambridge Cambridge United Kingdom

Vol. 15, No. 3, 2021