{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R647",
  "content_hash": null,
  "slug": "rdcr-claim-exact-prefix-maximum-fourteen",
  "type": "claim",
  "title": "The exact maximum through ten billion is fourteen",
  "summary": "A complete sweep of all ten billion divisor counts proves that no length-15 rainbow interval ends by 10,000,000,000.",
  "relevance": "For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-exact-prefix-maximum-fourteen (“The exact maximum through ten billion is fourteen”) records a bound, answer, status fact, or structural consequence. The record states: A complete sweep of all ten billion divisor counts proves that no length-15 rainbow interval ends by 10,000,000,000.",
  "relevance_source": "recorded",
  "body": "For a segment \\([a,b]\\), the program initializes a residual copy of every integer and a divisor count equal to 1. For each prime \\(p\\leq\\sqrt b\\), it removes the full power \\(p^e\\) from every multiple of \\(p\\) in the segment and multiplies the divisor count by \\(e+1\\). A residual factor greater than 1 is prime and contributes a final factor 2. This reconstructs\n\\[\n\\tau(n)=\\prod_{p^e\\parallel n}(e+1)\n\\]\nexactly for every integer in the segment.\n\nThe sliding search stores the most recent position of each divisor count. Immediately before processing \\(n\\), let \\(s\\) begin the current rainbow suffix. After reading \\(d=\\tau(n)\\), replace\n\\[\ns\\leftarrow\\max\\{s,\\operatorname{last}(d)+1\\}.\n\\]\nThen \\(n-s+1\\) is the longest rainbow interval ending at \\(n\\). Both \\(s\\) and the last-position table persist across segment boundaries. Maximizing this value over all \\(1\\leq n\\leq10^{10}\\) gives 14, first at the displayed witness. This proves the prefix maximum and rules out every length-15 interval ending in that checked range.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every interval of consecutive positive integers whose final term is at most 10000000000",
    "bounds": {
      "interval_end": {
        "min": 1,
        "max": 10000000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1510.07081",
      "locator": "rdcr-artifact-segmented-prefix-sweep"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1510.07081",
    "locator": "rdcr-artifact-segmented-prefix-sweep"
  },
  "relations": [
    {
      "slug": "R646",
      "title": "Segmented factor sieve and cross-segment rainbow search",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "rainbow-divisor-count-run-1e12",
      "title": "rainbow divisor count run 1e12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}