On a Conjecture of Kemnitz


-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.

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Received August 23, 1999

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Rónyai, L. On a Conjecture of Kemnitz. Combinatorica 20, 569–573 (2000). https://doi.org/10.1007/s004930070008

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  • AMS Subject Classification (1991) Classes:  11B50, 11P21