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 Zn 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 Zdn (the direct sum of d copies of Zn) there is a subsequence of length n whose sum is 0 in Zdn. 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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics