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The Erdős–Selfridge
problem with square-free moduli
Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba |
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Vol. 15 (2021), No. 3, 609–626
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Abstract |
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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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Keywords
covering systems, Erdős–Selfridge problem
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Mathematical Subject Classification 2010
Primary: 11B25
Secondary: 11A07, 11N35
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Milestones
Received: 31 January 2019
Revised: 14 August 2020
Accepted: 18 September 2020
Published: 20 May 2021
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Authors |
| Paul Balister | |
| University of Oxford Oxford United Kingdom |
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| Béla Bollobás | |
| University of Cambridge Cambridge United Kingdom |
University of Memphis Memphis, TN United States |
| Robert Morris | |
| IMPA Rio de Janeiro Brazil |
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| Julian Sahasrabudhe | |
| University of Cambridge Cambridge United Kingdom |
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| Marius Tiba | |
| University of Cambridge Cambridge United Kingdom |
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