Erdős (Erdos) Problem #326: status in 2026

Erdős Problem #326

Does there exist A = a_1 < a_2 < ⋯ ⊂ ℕ which is a minimal basis of order 2 (i.e. every large integer is the sum of 2 elements from A, and no proper subset of A has this property), such that lim_k→∞ a_k/k^2 = c for some c ≠ 0? Erdős and Graham conjectured a negative answer to this question [ErGr80].

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #326 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_326 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #326 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_326`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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-326,
  title        = {Erdős Problem #326},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-326}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Does there exist $A = {a_{1} < a_{2} < \dots} \subset N$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$ , and no proper subset of $A$ has this property), such that $k \to \infty lim \frac{a _{k}}{k ^{2}} = c$ for some $c \neq = 0$ ?

Erdős and Graham conjectured a negative answer to this question [ErGr80].

"Minimal basis of order $2$ " is formalised as Minimal for the predicate Set.IsAsymptoticAddBasisOfOrder · 2 on sets of naturals ordered by inclusion.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«326».

theorem erdos_326 : ∃ (a : ℕ → ℕ), StrictMono a ∧
    Minimal (fun A : Set ℕ ↦ A.IsAsymptoticAddBasisOfOrder 2) (Set.range a) ∧
      ∃ (c : ℝ), c ≠ 0 ∧ Tendsto (fun n ↦ (a n : ℝ) / n ^ 2) atTop (𝓝 c)

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/326. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/326
[ErGr80] Erdős, P. and Graham, R. L., *Old and new problems and results in combinatorial number theory*. Monographies de L'Enseignement Mathématique (1980), p. 47.
[Ca57] Cassels, J. W. S., Über Basen der natürlichen Zahlenreihe. Abh. Math. Sem. Univ. Hamburg (1957), 247-257.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.