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Zero-sumfree sequences in cyclic groups and some arithmetical application. (English) Zbl 1018.11011
A sequence of residues modulo $$n$$ is zero-sumfree if no nonempty subsequence has sum 0. It is easy to see that the maximal length of such a sequence is $$n-1$$, and equality occurs only if the same primitive element is repeated $$n-1$$ times. Here it is proved that even a sequence of $$[n/2+1]$$ elements contains primitive elements repeated $$m$$ times, where $$m=\lceil (n+5)/6 \rceil$$ if $$n$$ is odd, and $$m=3$$ if $$n$$ is even. This improves a result of W. Gao and A. Geroldinger [Combinatorica 18, 519-527 (1998; Zbl 0968.11016)]. It is shown by examples that this bound cannot be improved, and that a zero-sumfree sequence of $$n/2$$ elements may not contain a primitive element if $$n$$ is even but not a power of $$2$$, so in some aspects this result is best possible.
This result is shown to have an application to factorization in Krull monoids.

MSC:
 11B75 Other combinatorial number theory 20K01 Finite abelian groups 20M14 Commutative semigroups
Zbl 0968.11016
Full Text:
References:
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