A quantitative aspect of non-unique factorizations: the Narkiewicz constants II

Weidong Gao, Yuanlin Li, and Jiangtao Peng. "A quantitative aspect of non-unique factorizations: the Narkiewicz constants II." Colloquium Mathematicae 124.2 (2011): 205-218. <http://eudml.org/doc/284341>.

@article{WeidongGao2011,
abstract = {Let K be an algebraic number field with non-trivial class group G and $_\{K\}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_\{k\}(x)$ denote the number of non-zero principal ideals $a_\{K\}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_\{k\}(x)$ behaves, for x → ∞, asymptotically like $x(log x)^\{1/|G|-1\} (loglogx)^\{_\{k\}(G)\}$. In this article, it is proved that for every prime p, $₁(C_\{p\}⊕ C_\{p\}) = 2p$, and it is also proved that $₁ (C_\{mp\}⊕ C_\{mp\}) = 2mp$ if $₁ (C_\{m\}⊕ C_\{m\}) = 2m$ and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that $₁(C_\{mn\}⊕ C_\{mn\}) = 2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.},
author = {Weidong Gao, Yuanlin Li, Jiangtao Peng},
journal = {Colloquium Mathematicae},
keywords = {non-unique factorization; zero-sum sequences; combinatorial constants},
language = {eng},
number = {2},
pages = {205-218},
title = {A quantitative aspect of non-unique factorizations: the Narkiewicz constants II},
url = {http://eudml.org/doc/284341},
volume = {124},
year = {2011},
}

TY - JOUR
AU - Weidong Gao
AU - Yuanlin Li
AU - Jiangtao Peng
TI - A quantitative aspect of non-unique factorizations: the Narkiewicz constants II
JO - Colloquium Mathematicae
PY - 2011
VL - 124
IS - 2
SP - 205
EP - 218
AB - Let K be an algebraic number field with non-trivial class group G and $_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_{k}(x)$ denote the number of non-zero principal ideals $a_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_{k}(x)$ behaves, for x → ∞, asymptotically like $x(log x)^{1/|G|-1} (loglogx)^{_{k}(G)}$. In this article, it is proved that for every prime p, $₁(C_{p}⊕ C_{p}) = 2p$, and it is also proved that $₁ (C_{mp}⊕ C_{mp}) = 2mp$ if $₁ (C_{m}⊕ C_{m}) = 2m$ and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that $₁(C_{mn}⊕ C_{mn}) = 2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.
LA - eng
KW - non-unique factorization; zero-sum sequences; combinatorial constants
UR - http://eudml.org/doc/284341
ER -