Long zero-free sequences in finite cyclic groups

Details Long zero-free sequences in finite cyclic groups Long zero-free sequences in finite cyclic groups

2006-02-25

arxiv.org/abs/math/0602568v1

Mathematics

Combinatorics

13 pages

Scientific paper

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than $n/2$ in the additive group $\Zn/$ of integers modulo $n$. The main result states that for each zero-free sequence $(a_i)_{i=1}^\ell$ of length $\ell>n/2$ in $\Zn/$ there is an integer $g$ coprime to $n$ such that if $\bar{ga_i}$ denotes the least positive integer in the congruence class $ga_i$ (modulo $n$), then $\Sigma_{i=1}^\ell\bar{ga_i}

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