The board / number theory / #316
Erdős Problem #316
DISPROVED LEANdatabase updated 2025-09-02
The statement as posed is false. 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).
Is it true that if \(A \subseteq \mathbb{N}\setminus\{1\}\) is a finite set with \(\sum_{n \in A} \frac{1}{n} < 2\) then there is a partition \(A=A_1 \sqcup A_2\) such that \(\sum_{n \in A_i} \frac{1}{n} < 1\) for \(i=1,2\)? This is not true in general, as shown by Sándor [Sa97]. The minimal counterexample is \(\{2,3,4,5,6,7,10,11,13,14,15\}\), found by Tom Stobart. This was formalized in Lean by Mehta.
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/316.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/316
Attempts
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