{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R153",
  "content_hash": null,
  "slug": "cmi-claim-classical-cubic-formula",
  "type": "claim",
  "title": "Cubic value-set sizes were classified classically",
  "summary": "The general four-case formula specializes immediately to the candidate.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "For \\(g(x)=x^3+ax^2+bx+c\\) over \\(\\mathbf F_p\\), \\(p>3\\), the cited source states\n\\[\n|g(\\mathbf F_p)|=\n\\begin{cases}\np,&a^2-3b=0,\\ p\\equiv-1\\pmod3,\\\\\n(p+2)/3,&a^2-3b=0,\\ p\\equiv1\\pmod3,\\\\\n(2p-1)/3,&a^2-3b\\ne0,\\ p\\equiv-1\\pmod3,\\\\\n(2p+1)/3,&a^2-3b\\ne0,\\ p\\equiv1\\pmod3.\n\\end{cases}\n\\]\nThe paper attributes the cubic classification to work of R. Daublebsky von Sterneck in 1908 and Richard Kantor in 1915. Setting \\(a=0\\), \\(b=1\\), and \\(c=0\\) places \\(x^3+x\\) in the last two cases because \\(-3\\ne0\\) in the requested characteristics. The database candidate is therefore a known special case.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every monic cubic x cubed plus a x squared plus b x plus c over every prime field of characteristic greater than 3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
      "locator": "Brazelton, Harrington, Litman, and Wong, Journal of Number Theory 264 (2024), pages 1-2, cubic formula and historical attribution; references [Kan15] and [vS08]"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
    "locator": "Brazelton, Harrington, Litman, and Wong, Journal of Number Theory 264 (2024), pages 1-2, cubic formula and historical attribution; references [Kan15] and [vS08]"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R154",
      "title": "The image of x cubed plus x has the proposed size",
      "object_type": "claim",
      "relation": "generalizes",
      "direction": "outgoing"
    },
    {
      "slug": "cubic-map-image-size",
      "title": "cubic map image size",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}