{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R611",
  "content_hash": null,
  "slug": "ptm61-claim-primitive-field-model",
  "type": "claim",
  "title": "The stated polynomial gives a primitive degree-61 field model",
  "summary": "Rabin's irreducibility criterion and a Lucas-Lehmer certificate show that a root has order 2^61-1.",
  "relevance": "For Least trinomial multiple of a primitive degree-61 polynomial, record ptm61-claim-primitive-field-model (“The stated polynomial gives a primitive degree-61 field model”) records a bound, answer, status fact, or structural consequence. The record states: Rabin's irreducibility criterion and a Lucas-Lehmer certificate show that a root has order 2^61-1.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\nf=x^{61}+x^{45}+x^{32}+x^2+1.\n\\]\nThe exact replay computes \\(x^{2^{61}}\\bmod f=x\\) and\n\\[\n\\gcd(x^2+x,f)=1.\n\\]\nSince 61 is prime, these are the two conditions in the finite-field irreducibility criterion for a degree-61 polynomial over \\(\\mathbf F_2\\). Hence \\(f\\) is irreducible.\n\nThe same replay applies the Lucas-Lehmer test to\n\\[\nM=2^{61}-1=2305843009213693951\n\\]\nand obtains final residue zero, certifying that \\(M\\) is prime. A root \\(\\alpha\\) of \\(f\\) is nonzero, and \\(f(1)=1\\), so \\(\\alpha\\neq1\\). Its multiplicative order divides the prime \\(M\\) and exceeds one. Therefore \\(\\alpha\\) has order \\(M\\), and \\(f\\) is primitive.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the polynomial x^61+x^45+x^32+x^2+1 over F_2 and its degree-61 quotient field",
    "bounds": {
      "polynomial_degree": {
        "min": 61,
        "max": 61
      },
      "field_cardinality_exponent": {
        "min": 61,
        "max": 61
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1411.4024",
      "locator": "The field-root and Zech-logarithm formulation appears in Peterlongo, Sala, and Tinnirello, section 2; exact irreducibility and Lucas-Lehmer replay in ptm61-artifact-field-verification"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1411.4024",
    "locator": "The field-root and Zech-logarithm formulation appears in Peterlongo, Sala, and Tinnirello, section 2; exact irreducibility and Lucas-Lehmer replay in ptm61-artifact-field-verification"
  },
  "models": [],
  "relations": [
    {
      "slug": "R608",
      "title": "Exact irreducibility and Mersenne-primality replay",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R610",
      "title": "No trinomial multiple occurs through degree 2^28",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "primitive-degree61-trinomial-multiple",
      "title": "primitive degree61 trinomial multiple",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}