{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1197",
  "content_hash": null,
  "slug": "erdos-problem-137-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 137: Let $k \\geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \\geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\\cdots(n+k)$ is never a powerful number?",
  "relevance": "For Powerful Numbers in Products of Consecutive Integers, record erdos-problem-137-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 137: Let $k \\geq 3$ be an integer and let $n$ be a positive integer.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 137: Let $k \\geq 3$ be an integer and let $n$ be a positive integer. A positive integer $N$ is called powerful if for every prime $p$ dividing $N$, we have $p^2$ also divides $N$. Consider the product of $k$ consecutive integers starting from $n+1$, that is, the product $(n+1)(n+2)\\cdots(n+k)$. Must there always exist a prime $p$ dividing this product such that $p^2$ does not divide it? Equivalently, is it true that for all $k \\geq 3$ and all $n$, the product of $k$ consecutive integers $(n+1)(n+2)\\cdots(n+k)$ is never a powerful number? 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/137",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/137",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1198",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-137",
      "title": "erdos problem 137",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}