The Erdős–Selfridge problem with square-free moduli
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 notoriou...
| Main Authors: | , , , , |
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| Format: | Journal article |
| Language: | English |
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Mathematical Sciences Publishers
2021
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| _version_ | 1797063832614469632 |
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| author | Balister, P Bollobás, B Morris, R Sahasrabudhe, J Tiba, M |
| author_facet | Balister, P Bollobás, B Morris, R Sahasrabudhe, J Tiba, M |
| author_sort | Balister, P |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T21:05:34Z |
| format | Journal article |
| id | oxford-uuid:3c55267c-60be-4652-89a7-db3db5f62c25 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:05:34Z |
| publishDate | 2021 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:3c55267c-60be-4652-89a7-db3db5f62c252022-03-26T14:13:06ZThe Erdős–Selfridge problem with square-free moduliJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3c55267c-60be-4652-89a7-db3db5f62c25EnglishSymplectic ElementsMathematical Sciences Publishers2021Balister, PBollobás, BMorris, RSahasrabudhe, JTiba, MA 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. |
| spellingShingle | Balister, P Bollobás, B Morris, R Sahasrabudhe, J Tiba, M The Erdős–Selfridge problem with square-free moduli |
| title | The Erdős–Selfridge problem with square-free moduli |
| title_full | The Erdős–Selfridge problem with square-free moduli |
| title_fullStr | The Erdős–Selfridge problem with square-free moduli |
| title_full_unstemmed | The Erdős–Selfridge problem with square-free moduli |
| title_short | The Erdős–Selfridge problem with square-free moduli |
| title_sort | erdos selfridge problem with square free moduli |
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