{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R417",
  "content_hash": null,
  "slug": "iqfd-claim-known-third-iterate-factorization",
  "type": "claim",
  "title": "The third iterate factorization is published",
  "summary": "Ahmadi and Monsef-Shokri give a factorization of f^[3] into two quartics while proving instability of binary trinomials.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Theorem 7 treats even-degree trinomials over \\(\\mathbb F_2\\). Its first case is the present polynomial. The authors write\n\\[\nf^{[3]}(x)=x^8+x^4+x^2+x+1\n=(x^4+x^3+1)(x^4+x^3+x^2+x+1).\n\\]\nBoth factors have degree 4, agreeing with the formula at \\(n=3\\). Their theorem establishes reducibility of the third iterate for the larger trinomial family. It does not state the common factor degree for every later iterate.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the third compositional iterate of f(x)=x^2+x+1 over F_2",
    "bounds": {
      "n": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1810.03142",
      "locator": "Ahmadi and Monsef-Shokri, A note on the stability of trinomials over finite fields, Theorem 7 and its n=1, s=1 case on page 6; Finite Fields and Their Applications 63 (2020), article 101649, DOI 10.1016/j.ffa.2020.101649"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1810.03142",
    "locator": "Ahmadi and Monsef-Shokri, A note on the stability of trinomials over finite fields, Theorem 7 and its n=1, s=1 case on page 6; Finite Fields and Their Applications 63 (2020), article 101649, DOI 10.1016/j.ffa.2020.101649"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R416",
      "title": "Every factor has the predicted degree",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R415",
      "title": "Exact factorization sweep through the degree-4096 iterate",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "iterated-quadratic-factor-degree",
      "title": "iterated quadratic factor degree",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}