{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R633",
  "content_hash": null,
  "slug": "qorp-claim-first-length-5001-prime",
  "type": "claim",
  "title": "The first qualifying prime is 1229237",
  "summary": "Modulo 1229237, the orbit has 5145 distinct residues, with preperiod 2520 and period 2625.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Write \\(x_0=0\\) and \\(x_{k+1}=x_k^2+1\\pmod p\\). An exhaustive increasing-prime sweep found that\n\\[\np=1{,}229{,}237\n\\]\nis the first prime whose orbit contains more than 5000 distinct residues. The values \\(x_0,\\ldots,x_{5144}\\) are distinct, and the first collision is\n\\[\nx_{5145}=x_{2520}=817978.\n\\]\nThus the preperiod is 2520, the period is 2625, and their sum is 5145. The 94,997 smaller primes all have total orbit length at most 5000. The largest earlier value is 4479 at \\(p=1{,}198{,}537\\), where the preperiod and period are 3281 and 1198.\n\nPrimality has two independent checks. Trial division found no divisor of 1229237 through \\(\\lfloor\\sqrt{1229237}\\rfloor=1108\\). A Pocklington certificate uses\n\\[\np-1=2^2\\cdot17\\cdot18077.\n\\]\nThe numbers 17 and 18077 are prime by trial division through 4 and 134. With base 2, one has \\(2^{p-1}\\equiv1\\pmod p\\). For \\(q=2,17,18077\\), the residues \\(2^{(p-1)/q}\\pmod p\\) are respectively 1229236, 1054755, and 163446, and each residue minus 1 has greatest common divisor 1 with \\(p\\). This is a complete Pocklington certificate.\n\nThe compact-JSON serialization of the 5145 distinct residues has SHA-256 digest `9cd1f30ec1c06adb76c1c437bccaabb81a6f023f65ca1b81f459704b85f51db3`.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every prime modulus p from 2 through 1229237 for the orbit of 0 under x -> x^2+1",
    "bounds": {
      "p": {
        "min": 2,
        "max": 1229237
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1701.02707",
      "locator": "Exact exhaustive C sweep and independent Python verification performed by TheoremDB entry research on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1701.02707",
    "locator": "Exact exhaustive C sweep and independent Python verification performed by TheoremDB entry research on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R631",
      "title": "Exhaustive increasing-prime sweep",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R632",
      "title": "Independent cycle and primality verification",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "quadratic-orbit-record-prime-10m",
      "title": "quadratic orbit record prime 10m",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}