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

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 $\equiv 0$ $(\mathrm{mod}n)$ 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.