Christian Reiher

A proof of the theorem according to which every prime number possesses property B

University of Rostock, Institute of Mathematics, 2010

https://doi.org/10.18453/rosdok_id00005826

Abstract: In the present thesis, we prove a conjecture belonging to a subbranch of additive combinatorics that is called combinatorial zero-sum theory. Denoting by p an arbitrary prime number, it has been known for about forty years that every sequence (P1, P2,..., P2p-1) consisting of 2p-1points from the affine plane Fp^2 possesses a non-empty subsequence the sum of whose elements equals zero. This fact has been first shown independently by Kruyswijk and Olson and is nowadays known to be an easy consequence of Alon´s combinatorial Nullstellensatz. A less obvious question asksfor a classification of all those sequences (P1, P2,..., P2p-2) of lenght 2p-2 whose only zero-sum subsequence is the empty one. This problem has been investigated by several researchers and in this respect a certain conjecture implying in particular that any such sequence contains p-2 equal points has attracted a great deal of attention in recent years. A precise version of this conjecture will be given below. By definition, the prime number p has property B if and only if it behaevs in accordance with the conjecture under discussion; by developing for the first time a powerful method for tackling such inverse problems over Fp^2, we prove the result alluded to in the title.

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