### Abstract

A classic theorem of Erdos, Ginzburg and Ziv states that in a sequence of 2n - 1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_{n} stand for the additive group of integers modulo n. Let s(n,d) denote the smallest integer s such that in any sequence of s elements from Z^{d}_{n} (the direct sum of d copies of Z_{n}) there is a subsequence of length n whose sum is 0 in Z^{d}_{n}. Kemnitz conjectured that s(n,2) = 4n - 3. In this note we prove that s(p,2) ≤ 4p - 2 holds for every prime p. This implies that the value of s(p,2) is either 4p - 3 or 4p - 2. For an arbitrary positive integer n it follows that s(n,2) ≤ (41/10)n. The proof uses an algebraic approach.

Original language | English |
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Pages (from-to) | 569-573 |

Number of pages | 5 |

Journal | Combinatorica |

Volume | 20 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 2000 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*20*(4), 569-573. https://doi.org/10.1007/s004930070008