Erdős (Erdos) Problem #954: status in 2026

Erdős Problem #954

Let 0=a_0<a_1<a_2<⋯ be the sequence of integers defined by a_0=0 and a_1=1, and a_k+1 is the smallest integer n for which the number of solutions to a_i+a_j ≤ n (with 0≤ i≤ j≤ k and j≥ 1) is <n. Is the number of solutions to a_i+a_j ≤ x equal to x+O(x^1/4+o(1))?

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 #954 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_954 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #954 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_954`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 0 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-954,
  title        = {Erdős Problem #954},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-954}},
  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

Nobody has worked on this problem here yet

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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

Let $0 = a_{0} < a_{1} < a_{2} < \dots$ be the sequence of integers defined by $a_{0} = 0$ and $a_{1} = 1$ , and $a_{k + 1}$ is the smallest integer $n$ for which the number of solutions to $a_{i} + a_{j} \leq n$ (with $0 \leq i \leq j \leq k$ and $j \geq 1$ ) is $< n$ .

Is the number of solutions to $a_{i} + a_{j} \leq x$ equal to $x + O (x^{1/4 + o (1)})$ ?

The sequence was constructed by Rosen [Er77c, p. 71]. We read $O (x^{1/4 + o (1)})$ as $O (x^{1/4 + ε})$ for every $ε > 0$ , and use the version with $a_{0} = 0$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«954».

theorem erdos_954 :
    ∀ ε > (0 : ℝ), (fun x : ℕ ↦ (solutionCount x : ℝ) - x) =O[atTop]
      (fun x : ℕ ↦ (x : ℝ) ^ ((1 : ℝ) / 4 + ε))

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/954. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_954.variants.upper — Erdős and Rosen could not even prove that the number of solutions to a_i+a_j≤ x is at most (1+o(1))x [Er77c, p. 71].

References

erdosproblems.com/954
[Er77c] Erdős, P., Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Math. 626, Springer (1977), 43-72. PDF

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.