### On covering numbers [chapter]

Zhi-Wei Sun
Combinatorial Number Theory
A positive integer n is called a covering number if there are some distinct divisors n 1 , . . . , n k of n greater than one and some integers a 1 , . . . , a k such that Z is the union of the residue classes a 1 (mod n 1 ), . . . , a k (mod n k ). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (primitive) covering number; in particular, we show that for any r = 2, 3, . . . there are
more » ... finitely many primitive covering numbers having exactly r distinct prime divisors. In 1980 P. Erdős asked whether there are infinitely many positive integers n such that among the subsets of D n = {d 2 : d | n} only D n can be the set of all the moduli in a cover of Z with distinct moduli; we answer this question affirmatively. We also conjecture that any primitive covering number must have a prime factorization p α 1 1 · · · p α r r (with p 1 , . . . , p r in a suitable order) which satisfies 0<t<s (α t + 1) p s − 1 for each 1 s r, with strict inequality when s = r. -Dedicated to Prof. R. L. Graham for his 70th birthday 1