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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
Citations:
Zbl 0968.11016
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References:
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