{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1218",
  "content_hash": null,
  "slug": "erdos-problem-18-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 18 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A positive integer $n$ is called practical if every integer $m$ with $1 \\leq m \\leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \\leq m \\leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\\log \\log m)^C$.",
  "relevance": "For erdos problem 18, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 18: A.",
  "relevance_source": "recorded",
  "body": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 18 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 18: A positive integer $n$ is called practical if every integer $m$ with $1 \\leq m \\leq n$ can be written as a sum of distinct positive divisors of $n$. For a practical number $n$, define $h(n)$ to be the maximum, over all integers $m$ with $1 \\leq m \\leq n$, of the minimum number of distinct divisors of $n$ needed to represent $m$ as their sum. Determine whether there exists a real constant $C > 0$ such that there exist infinitely many practical numbers $m$ satisfying $h(m) < (\\log \\log m)^C$.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/18",
      "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/18",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1217",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "erdos-problem-18",
      "title": "erdos problem 18",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}