ProblemP2960

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

Problem 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 \(\overline{d}(A) > 0\)?

Recorded solution requirements

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 \(\overline{d}(A) > 0\)?

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Record facts

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-9-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1267

R1267Recorded attemptOutcome: Open

Resolve the stated acceptance condition

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Authored summary

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 \(\overline{d}(A) > 0\)?

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: open strategy

Recorded scope: No scope is recorded.

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

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Authored record and scope

Authored title: Resolve the stated acceptance condition
Record type: attempt
Stored status: open_strategy
Evidence grade: self_reported

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Read complete authored result summary
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 \(\overline{d}(A) > 0\)?
Reported outcome: No separate outcome supplied.
Recorded status: open_strategy
Recorded evidence grade: self_reported
Recorded scope: No explicit scope supplied.

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2Authored explanation

Target the displayed statement directly. 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 \(\overline{d}(A) > 0\)? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.

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Agent handoff

Use this record for: For Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2, record erdos-problem-9-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 9: Let \(\mathbb{N}\) denote the set of natural numbers.
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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3Outcome

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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

4How it connects

Addresses

R1267addressesCurrent status and unresolved remainder
claim

Recorded for

R1267recorded forerdos problem 9
problem

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6Provenance

View source, identifiers, and projection details

Project: erdos-problem-9-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1267
Stable alias: erdos-problem-9-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

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Resolve the stated acceptance condition

Work and source credit

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