The board / number theory / #858
Erdős Problem #858
SOLVEDdatabase updated 2026-04-24
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` has no solution to \(at=b\) with \(a,b\in A\) and the smallest prime factor of \(t\) greater than \(a\).
The maximum of \(\sum_{n\in A}\frac{1}{n}\) over all admissible \(A\subseteq \{1,\ldots,N\}\).
Let \(A\subseteq \{1,\ldots,N\}\) be such that there is no solution to \(at=b\) with \(a,b\in A\) and the smallest prime factor of \(t\) is \(>a\). Estimate the maximum of \[\frac{1}{\log N}\sum_{n\in A}\frac{1}{n}.\] This has been solved by Chojecki and GPT-5.4 Pro, who show that for large \(N\) \[\max_A \sum_{n\in A}\frac{1}{n}=(c+o(1))\log N\] where the maximum is over all \(A\subseteq \{1,\ldots,N\}\) with the stated property and \(c\approx 0.618\cdots\) is an explicit constant.
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/858.Prizenone
Topicsnumber theory, primitive sets
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/858
Attempts
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