Erdős Problem #14 — open problem in number theory

Erdős Problem #14

Let A ⊆ ℕ. Let B ⊆ ℕ be the set of integers which are representable in exactly one way as the sum of two elements from A. Is it true that for all ε > 0 and large N, |1,…,N ∖ B| ≫_ε N^1/2 - ε?

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-14,
  title        = {Erdős Problem #14},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-14}},
  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_14.parts.i. Let $A \subseteq N$ . Let $B \subseteq N$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$ . Is it true that for all $ϵ > 0$ and large $N$ , $∣ {1, \dots, N} ∖ B ∣ ≫_{ϵ} N^{1/2 - ϵ}$ ?

erdos_14.parts.ii. Is it possible that $∣ {1, \dots, N} ∖ B ∣ = o (N^{\frac{1}{2}})$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«14» (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_14.parts.i :
    answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε

theorem erdos_14.parts.ii :
    answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot

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/14. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/14

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.