On A Combinatorial Problem Connected withFactorizations
A quantitative aspect of non-unique factorizations: the Narkiewicz constants III
Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves for x → ∞ asymptotically like . We prove, among other results, that for all integers n₁,n₂ with 1 < n₁|n₂.
A quantitative aspect of non-unique factorizations: the Narkiewicz constants II
Let K be an algebraic number field with non-trivial class group G and be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let denote the number of non-zero principal ideals with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that behaves, for x → ∞, asymptotically like . In this article, it is proved that for every prime p, , and it is also proved that if and m is large enough. In particular, it is shown that for...
On additive bases II
Let G be an additive finite abelian group, and let S be a sequence over G. We say that S is regular if for every proper subgroup H ⊆ G, S contains at most |H|-1 terms from H. Let ₀(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., every element of G can be expressed as the sum over a nonempty subsequence of S. The constant ₀(G) has been determined previously only for the elementary abelian groups. In this paper, we determine...
Inverse zero-sum problems {III}
On the index of sequences over cyclic groups
Inverse zero-sum problems
On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths
Let G be an additive finite abelian group. For every positive integer ℓ, let be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine for certain finite groups, including cyclic groups, the groups and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences...
On zero-sum sequences in .
Subsums of a zero-sum free subset of an abelian group.
On subsequence sums of a zero-sum free sequence. II.
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