{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1267",
  "content_hash": null,
  "slug": "erdos-problem-9-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "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\n\\[ 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 \\,\\}. \\]\nThe set \\(A\\) is known to be infinite. For a subset \\(S \\subseteq \\mathbb{N}\\), the upper density of \\(S\\) is defined as\n\\[ \\overline{d}(S) = \\limsup_{N \\to \\infty} \\frac{|S \\cap \\{1, 2, \\ldots, N\\}|}{N}. \\]\nDoes \\(A\\) have positive upper density, i.e., is \\(\\overline{d}(A) > 0\\)?",
  "relevance": "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.",
  "relevance_source": "recorded",
  "body": "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\n\\[ 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 \\,\\}. \\]\nThe set \\(A\\) is known to be infinite. For a subset \\(S \\subseteq \\mathbb{N}\\), the upper density of \\(S\\) is defined as\n\\[ \\overline{d}(S) = \\limsup_{N \\to \\infty} \\frac{|S \\cap \\{1, 2, \\ldots, N\\}|}{N}. \\]\nDoes \\(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.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/9",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/9",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1268",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-9",
      "title": "erdos problem 9",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}