On Covering Numbers

A positive integer n is called a covering number if there are some distinct divisors n1, . . . , nk of n greater than one and some integers a1, . . . , ak such that Z is the union of the residue classes a1(mod n1), . . . , ak(mod nk). 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 infinitely 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 Dn = {d ! 2 : d | n} only Dn 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 p1 1 · · · pαr r (with p1, . . . , pr in a suitable order) which satisfies ∏ 0<t<s(αt + 1) ! ps − 1 for each 1 " s " r, with strict inequality when s = r. –Dedicated to Prof. R. L. Graham for his 70th birthday