Problem 289

Is it true that, for all sufficiently large $k$ , there exist finite intervals $I_1,\ldots,I_k\subset \mathbb{N}$ , distinct, not overlapping or adjacent, with $\lvert I_i\rvert \geq 2$ for $1\leq i\leq k$ such that

1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?

Source: erdosproblems.com/289

Number theory
Unit fractions

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

Open on the site: the label is OPEN (page last edited 22 September 2025; accessed 2026-10-07), and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is claimed, claim proved: four full proof claims are pending, three on the site's proof-claim tab, of 4 September, 7 September and 26 September 2026 (Land, Tang, Budden), each declaring AI assistance and each asserting a form stronger than the restricted statement, and a Lean 4 proof of the statement by the LEAP prover agent, posted 1 October 2026 and linked by formal-conjectures since 7 October 2026 (LEAP); none is accepted by the site, a referee or a named mathematician, and no build or review of any of them is recorded. No proof, disproof or accepted resolution of the restricted statement was found in the search whose scope the Current assessment records. The unrestricted variant, in which intervals may repeat or overlap, is settled affirmatively by an argument in the site's discussion (Kovač, September 2025) that the site's commentary adopts.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 34 (the site gives no page number). Library home: Erdős and Graham 1980.
Hah78: Hahn, L.-S., Problem E2689. Amer. Math. Monthly 85 (1978), p. 47 (the monograph's [Hah (78)]). Not held in the library; the problem's text is reprinted at the head of the solution [Mon79], p. 224.
Mon79: Montgomery, P., Solution to Problem E2689. Amer. Math. Monthly 86 (1979), no. 3, 224 (the monograph's [Mon (79)]; DOI 10.2307/2321534; the site credits the solution to Hickerson and Montgomery). The reprinted problem, the two sets and the note on Hickerson are on p. 224. Library home: Montgomery 1979. - The 1979 advance chapter of the monograph (library card) is the van der Waerden chapter and contains no unit-fraction passage; the passage is in the 1980 monograph.

Problem 289

Status

Proof claims

Discussion