The board / number theory / #300
Erdős Problem #300
SOLVED LEANdatabase updated 2026-08-24
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. 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).
\(A(N)\) is the maximal cardinality of \(A\subseteq \{1,\ldots,N\}\) such that \(\sum_{n\in S}\frac{1}{n}\neq 1\) for all \(S\subseteq A\).
Let \(A(N)\) denote the maximal cardinality of \(A\subseteq \{1,\ldots,N\}\) such that \(\sum_{n\in S}\frac{1}{n}\neq 1\) for all \(S\subseteq A\). Estimate \(A(N)\). Erdős and Graham [ErGr80] believe the answer is \(A(N)=(1+o(1))N\). Croot [Cr03] disproved this, showing the existence of some constant \(c<1\) such that \(A(N)<cN\) for all large \(N\). It is trivial that \(A(N)\geq (1-\frac{1}{e}+o(1))N\). Liu and Sawhney [LiSa24] have proved that \(A(N)=(1-1/e+o(1))N\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/300.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA390393
Original pageerdosproblems.com/300
Attempts
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