The board / number theory / #298
Erdős Problem #298
PROVED LEANdatabase updated 2025-12-29
The statement as posed is true, and a proof exists. 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).
Does every set \(A \subseteq \mathbb{N}\) of positive density contain some finite \(S \subset A\) such that \(\sum_{n \in S} \frac{1}{n} = 1\)? The answer is yes, proved by Bloom [Bl21] (even if 'positive density' is interpreted as 'positive upper density', which is likely what Erdős intended). The theorem below uses the positive-upper-density interpretation; the literal natural-density interpretation is recorded separately. This was formalized in Lean 3 by Bloom and 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/298.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/298
Attempts
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