Erdős Problem #289 — open problem in number theory

Erdős Problem #289

Is it true that, for all sufficiently large k, there exist finite intervals I_1, dotsc, I_k ⊂ ℕ, distinct, not overlapping or adjacent, with |I_i| ≥ 2 for 1 ≤ i ≤ k such that 1 = Σ_i=1^k Σ_n ∈ I_i 1/n?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-289,
  title        = {Erdős Problem #289},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-289}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is it true that, for all sufficiently large $k$ , there exist finite intervals $I_{1}, \dots, I_{k} \subset N$ , distinct, not overlapping or adjacent, with $∣ I_{i} ∣ \geq 2$ for $1 \leq i \leq k$ such that $1 = i = 1 \sum k n \in I_{i} \sum \frac{1}{n} ?$ Here two intervals are adjacent if their union is again an interval, so any two of the $I_{i}$ must be separated by at least one integer.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«289». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_289 : answer(sorry) ↔
    (∀ᶠ k : ℕ in atTop, ∃ I : Fin k → ℕ × ℕ,
    (∀ i, (I i).1 < (I i).2) ∧
    (∀ i j, i ≠ j → (I i).2 + 1 < (I j).1 ∨ (I j).2 + 1 < (I i).1) ∧
    ∑ i, ∑ n ∈ .Icc (I i).1 (I i).2, (n⁻¹ : ℚ) = 1)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/289. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/289

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.