New upper bounds for the Davenport and for the Erdo{double acute}s-Ginzburg-Ziv constants

Abstract

Let G be a finite abelian group (written additively) of rank r with invariants n 1, n 2, . . ., n r, where n r is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ n r + n r-1 + (c(3) - 1)n r-2 + (c(4) - 1) n r-3 + · · · + (c(r) - 1)n 1 + 1, where c(i) is the Alon-Dubiner constant, which depends only on the rank of the group ℤ inr. Also, we shall give an application of Davenport's constant to smooth numbers related to the Quadratic sieve. © 2012 Springer Basel AG.