{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R154",
  "content_hash": null,
  "slug": "cmi-claim-exact-image-size",
  "type": "claim",
  "title": "The image of x cubed plus x has the proposed size",
  "summary": "For every prime p at least 5, the image has size floor((2p+1)/3).",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Let \\(V_p=\\{x^3+x:x\\in\\mathbf F_p\\}\\), and let \\(\\chi=\\left(\\frac{-3}{p}\\right)\\). The fiber calculation in the companion record gives\n\\[\n|V_p|=\\frac{2p+\\chi}{3}.\n\\]\nQuadratic reciprocity gives\n\\[\n\\left(\\frac{-3}{p}\\right)=\\left(\\frac{p}{3}\\right)=\n\\begin{cases}1,&p\\equiv1\\pmod3,\\\\-1,&p\\equiv2\\pmod3.\\end{cases}\n\\]\nConsequently\n\\[\n|V_p|=\n\\begin{cases}(2p+1)/3,&p\\equiv1\\pmod3,\\\\(2p-1)/3,&p\\equiv2\\pmod3,\n\\end{cases}\n=\\left\\lfloor\\frac{2p+1}{3}\\right\\rfloor.\n\\]\nThis proves the candidate for every requested prime. The result is also the case \\(a=0,b=1,c=0\\) of the classical formula for the value set of \\(x^3+ax^2+bx+c\\).",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every prime p at least 5"
  },
  "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, Residue sums of Dickson polynomials over finite fields, Journal of Number Theory 264 (2024), page 2 of the published article and page 1 of arXiv:2103.09119v2, displayed cubic value-set formula; DOI 10.1016/j.jnt.2024.04.016"
    },
    "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, Residue sums of Dickson polynomials over finite fields, Journal of Number Theory 264 (2024), page 2 of the published article and page 1 of arXiv:2103.09119v2, displayed cubic value-set formula; DOI 10.1016/j.jnt.2024.04.016"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R155",
      "title": "The complete fiber profile follows from one character sum",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R153",
      "title": "Cubic value-set sizes were classified classically",
      "object_type": "claim",
      "relation": "generalizes",
      "direction": "incoming"
    },
    {
      "slug": "R152",
      "title": "Exact image enumeration for 5,131 primes",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "cubic-map-image-size",
      "title": "cubic map image size",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}