The board / number theory / #311
Erdős Problem #311
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`delta N` is the minimal non-zero value of \(\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert\) as \(A\) ranges over all subsets of \(\{1,\ldots,N\}\). The empty set gives the value \(1\), so the set below is nonempty for every \(N\).
Let \(\delta(N)\) be the minimal non-zero value of \(\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert\) as \(A\) ranges over all subsets of \(\{1,\ldots,N\}\). Is it true that \[\delta(N)=e^{-(c+o(1))N}\] for some constant \(c\in (0,1)\)? The formulation in [ErGr80] has the additional condition that \(A\) contains no \(S\) with \(\sum_{n\in S}\frac{1}{n}=1\); Kovac (in the comments on the problem page) showed that the formulation above is equivalent. We state \(\delta(N)=e^{-(c+o(1))N}\) as \(-\log \delta(N) / N \to c\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/311.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/311
Attempts
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Uniform elementary equivalence proof, corrected rigorous 5/6 subsequence argument, and an exact independently checked table extended from N=25 to N=50; the asymptotic question remains open at the expl
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