Erdős Problem #288 — open problem in number theory

Erdős Problem #288

Is it true that there are only finitely many pairs of intervals I_1, I_2 such that Σ_n_1 ∈ I_1 1/n_1 + Σ_n_2 ∈ I_2 1/n_2 ∈ 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-288,
  title        = {Erdős Problem #288},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-288}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is it true that there are only finitely many pairs of intervals $I_{1}$ , $I_{2}$ such that $n_{1} \in I_{1} \sum \frac{1}{n _{1}} + n_{2} \in I_{2} \sum \frac{1}{n _{2}} \in N ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«288». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_288 : answer(sorry) ↔ Set.Finite { I : Fin 2 → ℕ+ × ℕ+ |
    ∀ j, (I j).1 ≤ (I j).2 ∧
      ∃ n : ℕ+, (∑ j : Fin 2, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n }

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/288. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_288.variants.i2_card_eq_1 — This is still open even if |I_2| = 1.
erdos_288.variants.k_intervals — It is perhaps true with two intervals replaced by any k intervals.
erdos_288.variants.exists_k_gt_2 — Is it true for any k > 2 that only finitely many k intervals satisfy this condition?

References

erdosproblems.com/288

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.