The board / number theory / #691
Erdős Problem #691
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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The set \(M_A = \{n \geq 1 : a \mid n \text{ for some } a \in A\}\) of positive multiples of elements of \(A\).
Given \(A \subseteq \mathbb{N}\), let \(M_A = \{n \geq 1 : a \mid n \text{ for some } a \in A\}\). Find a necessary and sufficient condition on \(A\) for \(M_A\) to have density \(1\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/691.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/691
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Classical finite-lcm reduction verified; a density-preserving blow-up gives a primitive gcd-1 reciprocal-divergent example of exact density 2/3, and all 1024 five-prime semiprime families were exhaust
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