Distinct length modular zero-sum subsequences: A proof of Graham's conjecture

TextLet S be a sequence of n   nonnegative integers not exceeding n−1n−1 such that S takes at least three distinct values. We show that S   has two nonempty (modn) zero-sum subsequences with distinct lengths. This proves a conjecture of R.L. Graham. The validity of this conjecture was verified by Erdős and Szemerédi for all sufficiently large prime n.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LftJj-E6aQA.