Problem Statement
Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in S}n\equiv 0\pmod{N}$?
Categories:
Number Theory
Progress
A conjecture of Erdős and Heilbronn [ErHe64]. The answer is yes. This was proved for $N$ prime by Olson [Ol68], and for all $N$ by Szemerédi [Sz70] (in fact for arbitrary finite abelian groups).Erdős speculated that perhaps the correct threshold is $(2N)^{1/2}$; this is also a conjecture of Selfridge, and has been proved when $N$ is prime by Balandraud [Ba12].
Hamidoune and Zémor [HaZe96] proved the bound $(1+o(1))\sqrt{2N}$ for arbitrary abelian groups of order $N$.
This is discussed in problem C15 of Guy's collection [Gu04].
Source: erdosproblems.com/540 | Last verified: January 15, 2026