Erdős Problem #32 — open problem in number theory

Erdős Problem #32

Does there exist a set A ⊆ ℕ such that |A ∩ 1, …, N| = o((log N)^2) and every sufficiently large integer can be written as p + a for some prime p and a ∈ A?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-32,
  title        = {Erdős Problem #32},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-32}},
  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

Does there exist a set $A \subseteq N$ such that $∣ A \cap {1, \dots, N} ∣ = o ((lo g N)^{2})$ and every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \in A$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«32». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_32 : answer(sorry) ↔ ∃ A : Set ℕ,
    IsAdditiveComplementToPrimes A ∧
    (fun N => (((Finset.Icc 1 N).filter (· ∈ A)).card : ℝ)) =o[atTop]
      fun N => (Real.log N) ^ 2

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/32. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_32.variants.log_bound — Can the bound O(log N) be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered \50 for the solution.

References

erdosproblems.com/32
[Erd54] Erdős, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954),

847-853.

[Guy04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437
[Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.

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.