{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1180",
  "content_hash": null,
  "slug": "erdos-problem-10-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \\cdots + 2^{a_m}$ where $p$ is a prime number, $m \\leq k$, and $a_1, a_2, \\ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\\mathcal{S}_k = \\mathbb{N} \\setminus \\{0, 1\\}$?",
  "relevance": "For erdos problem 10, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For.",
  "relevance_source": "recorded",
  "body": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement.\n\nA complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \\cdots + 2^{a_m}$ where $p$ is a prime number, $m \\leq k$, and $a_1, a_2, \\ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\\mathcal{S}_k = \\mathbb{N} \\setminus \\{0, 1\\}$?",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/10",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/10",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "relations": [
    {
      "slug": "R1179",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "erdos-problem-10",
      "title": "erdos problem 10",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}