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The claim. Let $n$ be a positive integer and $S$ a sequence of $n$ integers in $[0,n-1]$. If $S$ takes at least three distinct values, then $S$ has two nonempty subsequences whose sums are divisible by $n$ and whose lengths differ. Read contrapositively with $n=p$ and $S=(a_1,\ldots,a_p)$, the $a_i$ taken as integers in $[0,p-1]$: a nonempty index set $S'\subseteq[p]$ with $\sum_{i\in S'}a_i\equiv0\pmod p$ is a nonempty zero-sum subsequence of length $\lvert S'\rvert$, so if every such index set has size $r$, at most two distinct residues occur. This is the statement of Problem 541 for every prime $p$, the residue $0$ admitted, and the theorem gives it for every modulus. The source is W. Gao, Y. O. Hamidoune and G. Wang, Distinct length modular zero-sum subsequences: a proof of Graham's conjecture, J. Number Theory 130 (2010), no. 6, 1425--1431, DOI 10.1016/j.jnt.2009.11.012, first posted as arXiv:0902.4758v1 on 27 February 2009 with the theorem in its abstract (the date this page is named by), cited from an author preprint whose file metadata is dated January 2010, a later version than that arXiv posting, and paged as Theorem 1.1 of Gao, Hamidoune and Wang (2010). The proof assumes every nonempty zero-sum subsequence has the same length $r$ and splits on $r\ge n/2$, handled with a zero-sum subsequence of length at most the maximal multiplicity, and $r<n/2$, handled with the Savchev--Chen and Yuan structure theorem for long zero-sum-free sequences; the authors say that this use of a structure theorem keeps it from being the simple proof Erdős and Szemerédi had hoped for. The statement was checked clause by clause; the proof (pp. 4--8 of the preprint) was read for structure only, and the journal text was not compared.

Acceptance. Refereed: the paper appeared in the Journal of Number Theory; the issue is dated June 2010 in the Crossref record (2026-10-07). Reviewed: the site's curator, Thomas Bloom, who is independent of the authors, labels the problem PROVED (LEAN), and his commentary (page last edited 8 April 2026) credits the proof for every modulus, prime or not, to this paper, with the large-prime case credited to Erdős and Szemerédi. Semantic Scholar listed nineteen citing records on 2026-09-18, none disputing the theorem by its title. Nothing here is independently reviewed by this project.

Related claims. The large-prime case for nonzero residues is the accepted partial claim Erdős and Szemerédi 1976; Grynkiewicz's later proof for every finite abelian group is the accepted claim Grynkiewicz 2009; the Lean proof the site's (LEAN) suffix refers to is the accepted claim Alexeev's Lean proof of 2025.

Depends on. Nothing in this wiki: the theorem is proved within the paper, whose card is linked above.