The board / number theory / #131
Erdős Problem #131
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).
A finite set \(A\) of naturals is *non-dividing* if no \(a \in A\) divides the sum of the elements of any nonempty subset \(S \subseteq A \setminus \{a\}\).
\(F(N)\) is the maximal size of a non-dividing subset of \(\{1, \dots, N\}\).
Let \(F(N)\) be the maximal size of \(A \subseteq \{1, \dots, N\}\) such that no \(a \in A\) divides the sum of any distinct elements of \(A \setminus \{a\}\). Is it true that \(F(N) > N^{1/2 - o(1)}\)? The answer is no, by the bound of Pham and Zakharov [PhZa24] (`erdos_131.variants.pham_zakharov`).
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/131.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA068063
Original pageerdosproblems.com/131
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