Mathematics – Number Theory
Scientific paper
2006-01-01
Mathematics
Number Theory
11 pages, to appear in INTEGERS (a special issue in honor of R. L. Graham's 70th birthday)
Scientific paper
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 infinitely many primitive covering numbers having exactly r distinct prime divisors. In 1980 P. Erdos asked whether there are infinitely many positive integers n such that among the subsets of D_n={d>1: 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_1^{alpha_1}...p_r^{alpha_r} (with p_1,...,p_r in a suitable order) which satisfies $\prod_{0
No associations
LandOfFree
On covering numbers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On covering numbers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On covering numbers will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-41531