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

A finite mathematical diagram showing odd integers assembled from a prime and two powers of two. — A prime and two powers of two feeding an integer sum.

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

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.

2Problem setup

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 \(S\).

Remark 1. 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.

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

Current 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

Record	Kind	Assessment
	Result	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

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

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Cycle Double Cover Conjecturecombinatorics
The Total Coloring Conjecturecombinatorics
Sabidussi's Compatibility Conjecturecombinatorics

How to cite

TheoremDB contributors, “Positive Upper Density of Odd Numbers Not Representable as a Prime Plus Two Powers of 2,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-9

This page as plain text: erdos-problem-9.md

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.

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

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance