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

TL;DR

This paper proves Graham's conjecture that sequences with uniform zero-sum subsequence lengths over [0,n-1] have at most two distinct values, extending prior results for large primes.

Contribution

It provides a complete proof of Graham's conjecture, generalizing earlier results limited to large prime cases.

Findings

Sequences with uniform zero-sum subsequence lengths have at most two distinct values.

Confirmed Graham's conjecture for all positive integers, not just primes.

Extended the validity of previous prime-based results to all integers.

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 $\equiv 0$ $\pmod n$ has length $=r,$ then $S$ has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erd\H{o}s and E. Szemer\'edi shows the validity of this conjecture if $n$ is a large prime number.