Problem Statement
Let $a_1,\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty and\[\sum_{i\in S}a_i\equiv 0\pmod{p}\]then $\lvert S\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?
Categories:
Number Theory
Progress
A question of Graham. This was proved by Erdős and Szemerédi [ErSz76] for $p$ sufficiently large and by Gao, Hamidoune, and Wang [GHW10] for all moduli (not necessarily prime).Source: erdosproblems.com/541 | Last verified: January 15, 2026