{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1210",
  "content_hash": null,
  "slug": "erdos-problem-145-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 145 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 145: Let $s_1 < s_2 < \\cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \\geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \\leq x$. For each real number $\\alpha \\geq 0$, consider the quantity\n\\[\n\\frac{1}{x} \\sum_{n \\in A(x)} (s_{n+1} - s_n)^{\\alpha}\n\\]\nas $x \\to \\infty$. Does there exist, for every $\\alpha \\geq 0$, a real number $\\beta$ such that this quantity converges to $\\beta$ as $x \\to \\infty$? Equivalently, does the limit\n\\[\n\\lim_{x \\to \\infty} \\frac{1}{x} \\sum_{s_n \\leq x} (s_{n+1} - s_n)^{\\alpha}\n\\]\nexist for all $\\alpha \\geq 0$?",
  "relevance": "For erdos problem 145, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 145 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 145.",
  "relevance_source": "recorded",
  "body": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 145 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 145: Let $s_1 < s_2 < \\cdots$ denote the sequence of squarefree positive integers, enumerated in increasing order. For each real number $x \\geq 0$, let $A(x)$ be the set of indices $n$ such that $s_n \\leq x$. For each real number $\\alpha \\geq 0$, consider the quantity\n\\[\n\\frac{1}{x} \\sum_{n \\in A(x)} (s_{n+1} - s_n)^{\\alpha}\n\\]\nas $x \\to \\infty$. Does there exist, for every $\\alpha \\geq 0$, a real number $\\beta$ such that this quantity converges to $\\beta$ as $x \\to \\infty$? Equivalently, does the limit\n\\[\n\\lim_{x \\to \\infty} \\frac{1}{x} \\sum_{s_n \\leq x} (s_{n+1} - s_n)^{\\alpha}\n\\]\nexist for all $\\alpha \\geq 0$?",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/145",
      "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/145",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1209",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "erdos-problem-145",
      "title": "erdos problem 145",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}