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.

中文翻译：

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具有无平方模的 Erdős-Selfridge 问题

一个覆盖系统是算术级数，其工会是整数集的有限集合。Erdős 于 1950 年发起了对具有不同模量的覆盖系统的研究，在接下来的几十年中，对其性质提出了许多问题。由于Erdős，一个特别臭名昭著的问题是询问是否存在模量不同且全为奇数的覆盖系统。我们证明，如果另外假设模数是无平方的，那么必定存在偶数模数。