Distinct Lengths Modular Zero-sum Subsequences: A Proof of Graham's Conjecture

2009/02/27 by Gao, Weidong, Hamidoune, Y. O., Wang, Guoqing · 1 citation
#05E15 #11B34 #11B60 #20D60 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0902.4758

Abstract

Let n be a positive integer and let S be a sequence of n integers in the interval [0,n-1]. If there is an r such that any nonempty subsequence with sum ≡ 0 \pmod n has length =r, then S has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erdős and E. Szemerédi shows the validity of this conjecture if n is a large prime number.

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• The structure of sequences with zero-sum subsequences of the same length on finite abelian groups of rank two

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