The board / number theory / #541
Erdős Problem #541
PROVED LEANdatabase updated 2025-12-30
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).
Let \(a_1, \dots, a_p\) be (not necessarily distinct) residues modulo a prime \(p\), 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 \(|S| = r\). Must there be at most two distinct residues amongst the \(a_i\)? This was formalized in Lean by Alexeev using Aristotle and ChatGPT.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/541.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/541
Attempts
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