{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R257",
  "content_hash": null,
  "slug": "ffe-claim-linear-solutions-and-first-nonlinear",
  "type": "claim",
  "title": "Linear solutions come from the golden-ratio polynomial",
  "summary": "The scalar map f(x)=a x works exactly when a^2-a-1=0; p=29 has two further nonlinear solutions.",
  "relevance": "For Power-of-two solution counts for a finite-field functional equation, record ffe-claim-linear-solutions-and-first-nonlinear (“Linear solutions come from the golden-ratio polynomial”) records a bound, answer, status fact, or structural consequence. The record states: The scalar map f(x)=a x works exactly when a^2-a-1=0; p=29 has two further nonlinear solutions.",
  "relevance_source": "recorded",
  "body": "For \\(f_a(x)=ax\\), substitution gives\n\\[\na^2x=(a+1)x\n\\]\nfor every \\(x\\), so the necessary and sufficient condition is\n\\[\na^2-a-1=0.\n\\]\nAt \\(p=5\\), this polynomial has the repeated root \\(a=3\\), giving one linear solution. For odd \\(p\\neq5\\), it has two roots exactly when 5 is a quadratic residue modulo \\(p\\), equivalently when \\(p\\equiv1\\) or \\(4\\pmod5\\). It has no roots at \\(p=2\\).\n\nThe exact census gives \\(N(29)=4\\). Since \\(29\\equiv4\\pmod5\\), precisely two of these functions are scalar-linear. The other two are nonlinear. Every smaller prime has either zero solutions, the single solution at 5, or the two scalar-linear solutions. Thus \\(p=29\\) is the first prime with a nonlinear solution.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "all scalar-linear functions f(x)=a x over prime fields, together with the exact count at p=29"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
      "locator": "Direct substitution, quadratic reciprocity for 5, and the exact p=29 census"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
    "locator": "Direct substitution, quadratic reciprocity for 5, and the exact p=29 census"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R258",
      "title": "The power-of-two claim is verified below 500 and unresolved in general",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "fibonacci-functional-equation-prime-count",
      "title": "fibonacci functional equation prime count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}