{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1628",
  "content_hash": null,
  "slug": "legendres-conjecture-status-packet-quality-20260801",
  "type": "claim",
  "title": "Dated status and exact unresolved remainder",
  "summary": "Unresolved in this packet after the dated source check. Strongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub. Exact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime.",
  "relevance": "For Legendre's conjecture, this successor gives readable dated status prose and the exact remaining research boundary.",
  "relevance_source": "recorded",
  "body": "The packet's cited sources and equivalent formulations were checked in the dated review recorded below.\n\nStrongest checked result: Unconditionally, every consecutive-square interval contains a 4-almost-prime. Assuming RH, primes occur between consecutive powers x^(2+delta) and (x+1)^(2+delta) in the ranges stated by Chamberland and Straub.\n\nExact unresolved remainder: Prove that every interval (n^2,(n+1)^2) contains a prime.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.sciencedirect.com/science/article/pii/S0022314X25001702",
      "locator": "Abstract and introduction"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.sciencedirect.com/science/article/pii/S0022314X25001702",
    "locator": "Abstract and introduction"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1092",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "legendres-conjecture",
      "title": "legendres conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}