## 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:

  Factorization in integral domains. (Editeur Daniel D. Anderson). 189, Marcel Dekker, Inc., New York, 1997.  Bovey, J.D., Erdös, P., Niven, I., Conditions for zero sum modulo n. Canad. Math. Bull.18 (1975), 27-29. · Zbl 0314.10040  Chapman, S., Geroldinger, A., Krull domains and monoids, their sets of lengths and associated combinatorial problems. In Factorization in integral domains, 73-112, 189, Marcel Dekker, New York, 1997. · Zbl 0897.13001  Eggleton, R.B., Erdös, P., Two combinatorial problems in group theory. Acta Arith.21 (1972), 111-116. · Zbl 0248.20068  Geroldinger, A., On non-unique factorizations into irreducible elements II. Number theory, Vol. II (Budapest, 1987), 723-757, Colloq. Math. Soc. János Bolyai51, North-Holland, Amsterdam, 1990. · Zbl 0703.11057  Geroldinger, A., Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z.197 (1988), 505-529. · Zbl 0618.12002  Gao, W., Geroldinger, A., On the structure of zerofree sequences. Combinatorica18 (1998), 519-527. · Zbl 0968.11016  Gao, W., Geroldinger, A., On long minimal zero sequences in finite abelian groups. Period. Math. Hungar.38 (1999), 179-211. · Zbl 0980.11014  Gao, W., Geroldinger, A., Systems of sets of lengths II. Abh. Math. Sem. Univ. Hamburg70 (2000), 31-49. · Zbl 1036.11054  Moser, L., Scherk, P., Distinct elements in a set of sums. Amer. Math. Monthly62 (1955), 46-47.
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