{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R515",
  "content_hash": null,
  "slug": "mgpc-claim-maximal-divisor-10m-20m",
  "type": "claim",
  "title": "The maximal-divisor criterion certifies 40,066 primes between ten and twenty million",
  "summary": "An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny.",
  "relevance": "For Prime exceptions to connectivity of the Markoff graph, record mgpc-claim-maximal-divisor-10m-20m (“The maximal-divisor criterion certifies 40,066 primes between ten and twenty million”) records a bound, answer, status fact, or structural consequence. The record states: An exhaustive exact-integer scan of all 606,028 primes with 10,000,000 < p <= 20,000,000 proves G_p connected for 40,066 of them by the criterion of Eddy, Fuchs, Litman, Martin, and Tripeny.",
  "relevance_source": "recorded",
  "body": "For a positive integer \\(n\\) and threshold \\(d\\), let \\(\\mathcal M_d(n)\\) be the divisibility-maximal members of \\(\\{e:e\\mid n,\\ e\\leq d\\}\\), and put\n\\[\nM_d=|\\mathcal M_d(p-1)|+|\\mathcal M_d(p+1)|.\n\\]\nTheorem 1.5 of Eddy et al. proves connectivity when no divisor \\(d\\mid p-1\\) or \\(d\\mid p+1\\) lies in either of its two forbidden intervals. The linked artifact evaluates both strict interval tests by squared and cross-multiplied integer inequalities. It factors \\(p-1\\) and \\(p+1\\), constructs every divisor, and computes each \\(M_d\\) from the definition.\n\nAmong the 606,028 primes in \\(10{,}000{,}000<p\\leq20{,}000{,}000\\), exactly 40,066 pass the criterion. The remaining 565,962 primes are undecided by this test. The certified-prime list, in increasing order and joined with ASCII commas without a terminal newline, has SHA-256 digest `5d8bbf2907288957ca191107018ac5a85cb13d620a9c9f56bb0c576fb3215cc3`.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all prime parameters 10,000,000 < p <= 20,000,000 were tested; connectivity is asserted for the 40,066 criterion-success primes",
    "bounds": {
      "p": {
        "min": 10000019,
        "max": 19999999
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2308.07579",
      "locator": "Theorem 1.5 and Section 7, Data on Connectivity"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2308.07579",
    "locator": "Theorem 1.5 and Section 7, Data on Connectivity"
  },
  "models": [],
  "relations": [
    {
      "slug": "R514",
      "title": "Connectivity is proved below one million and beyond an explicit threshold",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R509",
      "title": "Exact maximal-divisor criterion scan",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R512",
      "title": "Dated source and convention audit",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "markoff-graph-prime-connectivity-exceptions",
      "title": "markoff graph prime connectivity exceptions",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}