On a Conjecture of Kemnitz

n

-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 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 (the direct sum of d copies of ) there is a subsequence of length n whose sum is 0 in . Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that 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 . The proof uses an algebraic approach.