Erdős Problem #786 — open problem in number theory

Erdős Problem #786

Let ε > 0. Is there some set A⊂ℕ of density > 1 - ε such that a_1⋯ a_r = b_1⋯ b_s with a_i, b_j∈ A can only hold when r = s?

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-786,
  title        = {Erdős Problem #786},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-786}},
  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

erdos_786.parts.i. Let $ϵ > 0$ . Is there some set $A \subset N$ of density $> 1 - ϵ$ such that $a_{1} \dots a_{r} = b_{1} \dots b_{s}$ with $a_{i}, b_{j} \in A$ can only hold when $r = s$ ?

erdos_786.parts.ii. Is there some set $A \subset {1, ..., N}$ of size $\geq (1 - o (1)) N$ such that $a_{1} \dots a_{r} = b_{1} \dots b_{s}$ with $a_{i}, b_{j} \in A$ can only hold when $r = s$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«786» (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_786.parts.i : answer(sorry) ↔ ∀ ε > 0, ε ≤ 1 →
    ∃ (A : Set ℕ) (δ : ℝ), 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ A.IsMulCardSet

theorem erdos_786.parts.ii : answer(sorry) ↔
    ∃ (A : ℕ → Set ℕ) (f : ℕ → ℝ) (_ : f =o[atTop] (1 : ℕ → ℝ)),
    ∀ N, A N ⊆ Set.Icc 1 (N + 1) ∧ (1 - f N) * N ≤ (A N).ncard ∧ (A N).IsMulCardSet

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/786. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/786

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.