The board / number theory / #317
Erdős Problem #317
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Is there some constant \(c>0\) such that for every \(n\geq 1\) there exists some \(\delta_k\in \{-1,0,1\}\) for \(1\leq k\leq n\) with \[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/317.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/317
Attempts
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Exact modular certificates prove the second inequality for every \(5\leq n\leq200000\) (and equality defeats it exactly at \(n=1,2,3,4\) in this range); the uniform prime-residue lemma and the first q
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