Let G be a finite abelian group (written additively) of rank r with invariants n1, n2, . . . , nr, where nr 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) ≤ nr + nr−1 + (c(3) − 1)nr−2 + (c(4) − 1)nr−3 + · · · + (c(r) − 1)n1 + 1, where c(i) is the Alon-Dubiner constant, which depends only on the rank of the group Z i nr . Also, we shall give an application of Davenport's constant to smooth numbers related to the Quadratic sieve.Mathematics Subject Classification (2010). Primary 11B75; Secondary 20K01.
Let [Formula: see text] be a finite abelian group with exponent exp[Formula: see text]. Let [Formula: see text]. The constant [Formula: see text] is defined as the least positive integer [Formula: see text] such that for any given sequence [Formula: see text] of elements of [Formula: see text] with length [Formula: see text] it has a [Formula: see text] length [Formula: see text]-weighted zero-sum subsequence. In this article, we obtain the exact value of [Formula: see text] for [Formula: see text] and an upper bound for the case [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] is an odd integer and [Formula: see text]. We also obtain the structural information on the extremal zero-sum free sequences.
For a finite abelian group [Formula: see text] with exponent [Formula: see text], let [Formula: see text]. The constant [Formula: see text] (respectively [Formula: see text]) is defined to be the least positive integer [Formula: see text] such that given any sequence [Formula: see text] over [Formula: see text] with length [Formula: see text] has a [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] (respectively at most [Formula: see text]). In [M. N. Chintamani and P. Paul, On some weighted zero-sum constants, Int. J. Number Theory 13(2) (2017) 301–308], we proved the exact value of this constant for the group [Formula: see text] and proved the structure theorem for the extremal sequences related to this constant. In this paper, we prove the similar results for the group [Formula: see text] and we obtained an upper bound when [Formula: see text] is replaced by any integer [Formula: see text].
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