ProblemP2986

The minimal upper density of a square-additive basis for the natural numbers

Problem statement

Erdős Problem 33: A set \(A \subseteq \mathbb{N}\) is called a square-additive basis if every natural number \(k\) can be written as \(k = a + n^2\) for some \(a \in A\) and some \(n \geq 0\). For a square-additive basis \(A\), define its upper square-root density as \[ \limsup_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{\sqrt{N}}. \] Determine the exact value of the infimum of the upper square-root density over all square-additive bases \(A \subseteq \mathbb{N}\).

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 33: A set \(A \subseteq \mathbb{N}\) is called a square-additive basis if every natural number \(k\) can be written as \(k = a + n^2\) for some \(a \in A\) and some \(n \geq 0\). For a square-additive basis \(A\), define its upper square-root density as \[ \limsup_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{\sqrt{N}}. \] Determine the exact value of the infimum of the upper square-root density over all square-additive bases \(A \subseteq \mathbb{N}\).

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Record facts

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-33-claim-status-20260731
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1232

R1232Sourced evidence

Current status and unresolved remainder

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Authored summary

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 33 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 33: A set \(A \subseteq \mathbb{N}\) is called a square-additive basis if every natural number \(k\) can be written as \(k = a + n^2\) for some \(a \in A\) and some \(n \geq 0\). For a square-additive basis \(A\), define its upper square-root density as \[ \limsup_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{\sqrt{N}}. \] Determine the exact value of the infimum of the upper square-root density over all square-additive bases \(A \subseteq \mathbb{N}\).

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Recorded status: reported

Recorded scope: No scope is recorded.

Originating problem: The minimal upper density of a square-additive basis for the natural numbers

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Authored record and scope

Authored title: Current status and unresolved remainder
Record type: claim
Stored status: reported
Evidence grade: sourced

2Authored explanation

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 33 as open. The unresolved remainder is the full displayed statement.

A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 33: A set \(A \subseteq \mathbb{N}\) is called a square-additive basis if every natural number \(k\) can be written as \(k = a + n^2\) for some \(a \in A\) and some \(n \geq 0\). For a square-additive basis \(A\), define its upper square-root density as \[ \limsup_{N \to \infty} \frac{|A \cap \{1, 2, \ldots, N\}|}{\sqrt{N}}. \] Determine the exact value of the infimum of the upper square-root density over all square-additive bases \(A \subseteq \mathbb{N}\).

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Agent handoff

Use this record for: For erdos problem 33, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 33 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 33: A.
Evidence boundary: The claim carries Supported evidence. Its source and scope define the supported boundary.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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Replay material: source only

3Evidence

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Verification source: www.erdosproblems.com ↗, See dataset.references[0] for the exact external source and locator.

4How it connects

Addressed by

Resolve the stated acceptance conditionaddressesR1232
attempt

Recorded for

R1232recorded forerdos problem 33
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Machine-readable record

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json

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  "summary": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 33 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 33: A set \\(A \\subseteq \\mathbb{N}\\) is called a square-additive basis if every natural number \\(k\\) can be written as \\(k = a + n^2\\) for some \\(a \\in A\\) and some \\(n \\geq 0\\). For a square-additive basis \\(A\\), define its upper square-root density as\n\\[\n\\limsup_{N \\to \\infty} \\frac{|A \\cap \\{1, 2, \\ldots, N\\}|}{\\sqrt{N}}.\n\\]\nDetermine the exact value of the infimum of the upper square-root density over all square-additive bases \\(A \\subseteq \\mathbb{N}\\).",
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6Provenance

View source, identifiers, and projection details

Project: erdos-problem-33-source-review
Locator: See dataset.references[0] for the exact external source and locator.
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1232
Stable alias: erdos-problem-33-claim-status-20260731
Projection: Reproduction fields are derived from the immutable record.

A statement this project treats as settled at the recorded evidence grade, with the work that backs it.

Current status and unresolved remainder

2Authored explanation

3Evidence

4How it connects

Addressed by

Recorded for

6Provenance

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