Zero-sum problem

In number theory, zero-sum problems are a certain class of combinatorial questions. In general, a finite abelian group G is considered. The zero-sum problem for the integer n is the following: Find the smallest integer k such that every sequence of elements of G with length k contains n terms that sum to 0.

In 1961 Paul Erdős, Abraham Ginzburg, and Abraham Ziv proved the general result for \mathbb{Z}/n\mathbb{Z} (the integers mod n) that

k = 2n - 1.\

Explicitly this says that any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n. This result is generally known as the EGZ theorem after its discoverers.

More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003[1]), and the weighted EGZ theorem (proved by David J. Grynkiewicz in 2005[2]).

References

  1. ^ Reiher, Christian (2007), "On Kemnitz' conjecture concerning lattice-points in the plane", The Ramanujan Journal 13 (1–3): 333–337, doi:10.1007/s11139-006-0256-y .
  2. ^ Grynkiewicz, D. J. (2006), "A Weighted Erdős-Ginzburg-Ziv Theorem", Combinatorica 26 (4): 445–453, doi:10.1007/s00493-006-0025-y .

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