The board / number theory / #542
Erdős Problem #542
SOLVEDdatabase updated 2025-08-31
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.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A set `A ⊆ {1, …, n}` such that `lcm(a, b) > n` for all distinct `a, b ∈ A`.
The integers `1 ≤ m ≤ n` which are not divisible by any element of `A`.
Is it true that if \(A\subseteq\{1,\ldots,n\}\) is a set such that \([a,b]>n\) for all \(a\neq b\), where \([a,b]\) is the least common multiple, then \[\sum_{a\in A}\frac{1}{a}\leq \frac{31}{30}?\] The answer is yes, proved by Schinzel and Szekeres [ScSz59]. The bound is best possible as \(A=\{2,3,5\}\) demonstrates.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 7 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/542.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/542
Attempts
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