Erdős Problem #354 — open problem in number theory

Erdős Problem #354

Let α,β∈ ℝ_>0 such that α/β is irrational. Is the multiset ⌊ α⌋,⌊ 2α⌋,⌊ 4α⌋,…∪ ⌊ β⌋,⌊ 2β⌋,⌊ 4β⌋,… complete?

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.

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Cite

@misc{cairn-erdos-354,
  title        = {Erdős Problem #354},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-354}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_354.parts.i. Let $α, β \in R_{> 0}$ such that $α / β$ is irrational. Is the multiset ${⌊ α ⌋, ⌊ 2 α ⌋, ⌊ 4 α ⌋, \dots} \cup {⌊ β ⌋, ⌊ 2 β ⌋, ⌊ 4 β ⌋, \dots}$ complete?

erdos_354.parts.ii. Let $α, β \in R_{> 0}$ such that $α / β$ is irrational. Is ${⌊ α ⌋, ⌊ γ α ⌋, ⌊ γ^{2} α ⌋, \dots} \cup {⌊ β ⌋, ⌊ γ β ⌋, ⌊ γ^{2} β ⌋, \dots}$ complete?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«354» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_354.parts.i : answer(sorry) ↔ ∀ᵉ (α > 0) (β > 0), Irrational (α / β) →
    IsAddCompleteNatSeq' (FloorMultiples.interleave α β 2)

theorem erdos_354.parts.ii : answer(sorry) ↔ ∃ γ ∈ Set.Ioo (1 : ℝ) 2, ∀ᵉ (α > 0) (β > 0), Irrational (α / β) →
    IsAddCompleteNatSeq' (FloorMultiples.interleave α β γ)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/354. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/354

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.