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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.

11B75 Other combinatorial number theory
20K01 Finite abelian groups
20M14 Commutative semigroups
Zbl 0968.11016
Full Text: DOI Numdam EuDML
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