# P2986: The minimal upper density of a square-additive basis for the natural numbers - ID: `P2986` - Reference: `erdos-problem-33` - Page: https://theoremdb.org/statements/P2986 - Record maturity: Reviewed problem with recorded work ## 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 \(A \subseteq \mathbb{N}\). ### Context 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. ### Problem setup - **Definition (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 (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 (The golden ratio).** The golden ratio is \(\varphi = (1 + \sqrt{5})/2\). - **Remark.** 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. ### What 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 \(A \subseteq \mathbb{N}\). ## 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. 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](#reference-1) ## Work ### Evidence for the current status **Claim 1 (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 \(A \subseteq \mathbb{N}\). 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 \(A \subseteq \mathbb{N}\). ### Background and intake notes - 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. ### Open directions - **Route 1** (reported): 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](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-33`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Erdős Problems database, Problem 33, maintained status record. Erdős Problems record 33, checked 2026-08-01. Problem 33; status field and linked bibliography https://www.erdosproblems.com/33 - 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 - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Records the current open status and links the literature attached to this exact numbered problem. - 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. 2. Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 33 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source 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. 3. Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 33. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/33.lean:L42; theorem erdos_33; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/33.lean#L42 - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31 - Source 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.