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Let $n$ be a positive integer and let $S$ be a sequence of $n$ integers inthe interval $[0,n-1]$. If there is an $r$ such that any nonempty subsequencewith sum $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most twodistinct values. This proves a conjecture of R. L. Graham. A previous result ofP. Erd\H{o}s and E. Szemer\'edi shows the validity of this conjecture if $n$ isa large prime number.

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