{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R143",
  "content_hash": null,
  "slug": "cgr127-claim-undirected-cayley-convention",
  "type": "claim",
  "title": "The candidate defines a simple undirected 42-regular graph",
  "summary": "The 42 nonzero cubes contain -1, so the connection set is closed under negation.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The group \\(\\mathbf F_{127}^{\\times}\\) is cyclic of order 126. If \\(g\\) is a generator, then the cube subgroup is\n\\[\nC=\\langle g^3\\rangle,\n\\qquad |C|=126/3=42.\n\\]\nAlso\n\\[\n-1=g^{63}=(g^{21})^3\\in C.\n\\]\nIt follows that \\(C=-C\\). Thus \\(x-y\\in C\\) if and only if \\(y-x\\in C\\), which settles the direction convention. Since zero is excluded, the graph has no loops. Translation gives 42 neighbors at every vertex, hence 2,667 edges.\n\nIn the terminology of Lim and Praeger this is the generalized Paley graph whose connection subgroup has order 42. Their definition requires the connection subgroup to have even order in odd characteristic, exactly the condition that makes the relation symmetric.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the nonzero cubic-residue connection set in F_127",
    "bounds": {
      "modulus": {
        "min": 127,
        "max": 127
      },
      "connection_set_size": {
        "min": 42,
        "max": 42
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/0605252",
      "locator": "Tian Khoon Lim and Cheryl E. Praeger, On generalised Paley graphs and their automorphism groups, Michigan Mathematical Journal 58 (2009), 293-308, Definition 1.1; the p=127 calculation is elementary"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0605252",
    "locator": "Tian Khoon Lim and Cheryl E. Praeger, On generalised Paley graphs and their automorphism groups, Michigan Mathematical Journal 58 (2009), 293-308, Definition 1.1; the p=127 calculation is elementary"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R141",
      "title": "The independence number is 11",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R140",
      "title": "Complete colored search excluding an independent 12-set",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "cubic-residue-graph-127-independence",
      "title": "cubic residue graph 127 independence",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}