The board / number theory / #319
Erdős Problem #319
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
What is the size of the largest \(A\subseteq\{1, \dots, N\}\) such that there is a function \(\delta : A \to \{-1, 1\}\) such that \[ \sum_{n\in A} \frac{\delta n}{n} = 0 \] and \[ \sum_{n\in A'}\frac{\delta n}{n} \neq 0 \] for all non-empty \(A'\subsetneq A\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/319.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/319
Attempts
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Live page status is OPEN, but it lists 1 claimed proof (and no current worker); the mandatory collision rule therefore forbids an attempt.
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