The board / number theory / #650
Erdős Problem #650
SOLVED LEANdatabase updated 2026-03-06
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. 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).
`f m` is the largest `r` such that whenever `A ⊆ {1, …, N}` has `m` elements, every interval in `[1, ∞)` of length `2 * N` contains `r` distinct integers `b 0, …, b (r - 1)`, each `b i` being divisible by some `a i ∈ A`, where `a 0, …, a (r - 1)` are distinct.
Let \(f(m)\) be such that if \(A\subseteq \{1,\ldots,N\}\) has \(\lvert A\rvert=m\) then every interval in \([1,\infty)\) of length \(2N\) contains \(\geq f(m)\) many distinct integers \(b_1,\ldots,b_r\) where each \(b_i\) is divisible by some \(a_i\in A\), where \(a_1,\ldots,a_r\) are distinct. Estimate \(f(m)\). GPT 5.4 Pro (prompted by He, Li, and Tang) proved \(f(m)\leq \lceil 2\sqrt{m}\rceil\). A corresponding lower bound was given by GPT 5.4 Pro and Aristotle; it is now known (see the paper of van Doorn, Li, and Tang [VLT26]) that \[f(m) = \min(m, \lceil 2\sqrt{m}\rceil)\] for all \(m\). This was formalized in Lean by van Doorn using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/650.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA027434
Original pageerdosproblems.com/650
Attempts
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