The board / number theory / #540
Erdős Problem #540
PROVED LEANdatabase updated 2026-04-15
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A finite set has a non-empty subset whose sum is zero.
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 \(\sum_{n\in S}n\equiv 0\pmod{N}\)? Szemerédi proved the answer is yes, in fact for arbitrary finite abelian groups.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/540.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA034463
Original pageerdosproblems.com/540
Attempts
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