[#P2986] The minimal upper density of a square-additive basis for the natural numbers

Problem. 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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1Context

This is Erdős Problem 33, concerning additive bases with respect to the squares. Erdős proved that this infimum is finite and strictly greater than 1. Wouter van Doorn established an upper bound of \(2\varphi^{5/2} \approx 6.66\), where \(\varphi = (1+\sqrt{5})/2\) is the golden ratio.

2Definitions

Definition 1 (A set \(A \subseteq \mathbb{N}\). A set \(A \subseteq \mathbb{N}\) is a square-additive basis if for every natural number \(k\), there exist \(a \in A\) and \(n \in \mathbb{N}\) such that \(k = a + n^2\).

Definition 2 (The upper square-root density of a set \(A \subseteq \mathbb{N}\). The upper square-root density of a set \(A \subseteq \mathbb{N}\) is \(\limsup_{N \to \infty} |A \cap \{1, 2, \ldots, N\}| / \sqrt{N}\), where \(|S|\) denotes the cardinality of a finite set \(S\).

Definition 3 (The golden ratio). The golden ratio is \(\varphi = (1 + \sqrt{5})/2\).

3What counts as a solution

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 \(A \subseteq \mathbb{N}\).

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). 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 \(A \subseteq \mathbb{N}\).^[1]

1Packet records

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Record	Kind	Assessment
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 \(A \subseteq \mathbb{N}\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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 \(A \subseteq \mathbb{N}\).^[1] Relevance to this problem 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 SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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 \(A \subseteq \mathbb{N}\). Open full record →
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 \(A \subseteq \mathbb{N}\).	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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 \(A \subseteq \mathbb{N}\).^[1] Relevance to this problem For The minimal upper density of a square-additive basis for the natural numbers, record erdos-problem-33-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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 \(n \geq 0\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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 \(A \subseteq \mathbb{N}\). Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. 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.

The maintained database entry for Erdős Problem 33 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

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

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“The minimal upper density of a square-additive basis for the natural numbers.” TheoremDB. P2986. Problem statement; statement text SHA-256 85e877ba2d2a04a0e394a9dc7f1f43426d6f7e25ec1c82b044d7674272b98633. https://theoremdb.org/statement/?ref=P2986

BibTeX

@misc{theoremdb-problem-85e877ba2d2a04a0e394a9dc7f1f43426d6f7e25ec1c82b044d7674272b98633,
  title = {{The minimal upper density of a square-additive basis for the natural numbers}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 85e877ba2d2a04a0e394a9dc7f1f43426d6f7e25ec1c82b044d7674272b98633},
  url = {https://theoremdb.org/statement/?ref=P2986}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

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1References

Packet source. Erdős Problems database, Problem 33, maintained status record. Erdős Problems record 33, checked 2026-08-01. Problem 33; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 33; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For The minimal upper density of a square-additive basis for the natural numbers: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 33 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For The minimal upper density of a square-additive basis for the natural numbers: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 33. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/33.lean:L42; theorem erdos_33; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For The minimal upper density of a square-additive basis for the natural numbers: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P2986] The minimal upper density of a square-additive basis for the natural numbers

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