# P2960: Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2

- ID: `P2960`
- Reference: `erdos-problem-9`
- Page: https://theoremdb.org/statements/P2960
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by
\[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \]
The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as
\[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \]
Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)?

### Context

This problem originates from Paul Erdős in combinatorial number theory. It concerns the distribution of odd integers that fail to be representable as the sum of a prime number and two (not necessarily distinct) powers of 2.

### Problem setup

- **Definition (A natural number).** A natural number is odd if it is not divisible by 2.
- **Definition (The upper density of a set \(S \subseteq \mathbb{N}\).** The upper density of a set \(S \subseteq \mathbb{N}\) is \(\limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}\), which measures the asymptotic proportion of natural numbers contained in \(S\).
- **Remark.** This problem originates from Paul Erdős in combinatorial number theory. It concerns the distribution of odd integers that fail to be representable as the sum of a prime number and two (not necessarily distinct) powers of 2.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by
\[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \]
The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as
\[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \]
Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 9 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by
\[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \]
The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as
\[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \]
Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)? [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 9 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by
\[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \]
The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as
\[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \]
Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 9 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 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by
\[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \]
The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as
\[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \]
Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 9 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 9 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 9: Let \(\mathbb{N}\) denote the set of natural numbers. Define a set \(A \subseteq \mathbb{N}\) by \[ A = \{\, n \in \mathbb{N} \mid n \text{ is odd and there do not exist } p, k, l \in \mathbb{N} \text{ such that } p \text{ is prime and } n = p + 2^k + 2^l \,\}. \] The set \(A\) is known to be infinite. For a subset \(S \subseteq \mathbb{N}\), the upper density of \(S\) is defined as \[ \overline{d}(S) = \limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}. \] Does \(A\) have positive upper density, i.e., is \(\overline{d}(A) > 0\)? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-9`, 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. <a id="reference-1"></a>Erdős Problems database, Problem 9, maintained status record. Erdős Problems record 9, checked 2026-08-01. Problem 9; status field and linked bibliography https://www.erdosproblems.com/9
   - Also cited at Problem 9; 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 Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2: Records the current open status and links the literature attached to this exact numbered problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 9 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 Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2: Pins the maintained database snapshot used for this release's dated status decision.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 9. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/9.lean:L73; theorem erdos_9; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/9.lean#L73
   - 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 Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2: Supplies the pinned formal declaration whose human-readable editorial statement is published here.