Irreducible Covering Sets A Solution of Erdős Problem 1189

Jeff Pickhardt

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Irreducible Covering Sets A Solution of Erdős Problem 1189

Abstract

A finite set of distinct integers 1 <n1 <···<nk is called a covering set if one can choose one residue class modulo each ni so that the resulting classes cover Z. It is irreducible if no proper subset is a covering set. We determine the asymptotic number of irreducible covering sets of size k, determine exactly the greatest possible largest modulus, show that the smallest possible largest modulus is k1+o(1), determine the order of magnitude of the greatest possible reciprocal sum, and deduce from a theorem of Sun that infinitely many integers have the property that their nontrivial divisors form an irreducible covering set.