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

Contents

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

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Definitions and notation

1Context

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.

2Definitions

Definition 1 (A natural number). A natural number is odd if it is not divisible by 2.

Definition 2 (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 \(S\).

3What 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 \(\overline{d}(A) > 0\)?

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 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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 \(\overline{d}(A) > 0\)?	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 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] Relevance to this problem For erdos problem 9, pins the dated research frontier: 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. 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 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 \(\overline{d}(A) > 0\)? Open full record →
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 \(\overline{d}(A) > 0\)?	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 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] Relevance to this problem 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 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 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. 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 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.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

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

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2.” TheoremDB. P2960. Problem statement; statement text SHA-256 998915f165579994b43999a22bf85983111a039378c2ac8d36e9e266b80218e3. https://theoremdb.org/statement/?ref=P2960

BibTeX

@misc{theoremdb-problem-998915f165579994b43999a22bf85983111a039378c2ac8d36e9e266b80218e3,
  title = {{Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 998915f165579994b43999a22bf85983111a039378c2ac8d36e9e266b80218e3},
  url = {https://theoremdb.org/statement/?ref=P2960}
}

Plain text: Built Markdown snapshot

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

1References

Packet source. Erdős Problems database, Problem 9, maintained status record. Erdős Problems record 9, checked 2026-08-01. Problem 9; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 9 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 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.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 9. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/9.lean:L73; theorem erdos_9; 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 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.

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

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

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