{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R768",
  "content_hash": null,
  "slug": "tc114-claim-min-coordinate-search-1e17",
  "type": "claim",
  "title": "The published complete-search frontier is 10^17 in the smallest coordinate",
  "summary": "The 2019 Charity Engine run searched all solutions with minimum absolute coordinate at most 10^17 and reported none for 114.",
  "relevance": "For A bounded three-cubes search for 114, record tc114-claim-min-coordinate-search-1e17 (“The published complete-search frontier is 10^17 in the smallest coordinate”) records a bound, answer, status fact, or structural consequence. The record states: The 2019 Charity Engine run searched all solutions with minimum absolute coordinate at most 10^17 and reported none for 114.",
  "relevance_source": "recorded",
  "body": "Booker and Sutherland's algorithm permutes a nonexceptional solution so that\n\\[\n|x|>|y|>|z|\n\\]\nand sets\n\\[\nd=|x+y|.\n\\]\nFor \\(|z|>\\sqrt{k}\\), every such solution has\n\\[\n0<d<(\\sqrt[3]{2}-1)|z|.\n\\]\nThe exceptional equal-absolute-value cases are handled separately. Consequently, using \\(z_{\\max}=B\\) and \\(d_{\\max}=(\\sqrt[3]{2}-1)B\\) searches every solution whose smallest absolute coordinate is at most \\(B\\).\n\nIn September 2019 the authors ran this complete parameter choice for all eleven then-unresolved targets, including 114, with\n\\[\nz_{\\max}=10^{17},\\qquad d_{\\max}=(\\sqrt[3]{2}-1)10^{17}.\n\\]\nThey report solutions for 42, 165, and 906. No solution for 114 appeared. This covers every triple in the smaller height box \\(H\\leq10^{17}\\), and it also covers every triple in the requested height box having at least one coordinate of absolute value at most \\(10^{17}\\).\n\nThe authors caution that their crowd-sourced searches may contain an undetected hardware or software error. This record therefore preserves the result as exhaustive published computational evidence rather than a formal impossibility proof.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "solutions of x^3+y^3+z^3=114 with min(abs(x), abs(y), abs(z)) at most 10^17",
    "bounds": {
      "minimum_absolute_coordinate": {
        "min": 0,
        "max": 100000000000000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1073/pnas.2022377118",
      "locator": "Booker and Sutherland, Sections 1 and 5.B, especially the algorithm coverage statement on pages 1-2, the September 2019 parameters on page 10, and Remark 5.1 on page 11"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1073/pnas.2022377118",
    "locator": "Booker and Sutherland, Sections 1 and 5.B, especially the algorithm coverage statement on pages 1-2, the September 2019 parameters on page 10, and Remark 5.1 on page 11"
  },
  "relations": [
    {
      "slug": "R767",
      "title": "The height-10^20 question remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "three-cubes-114-height-1e20",
      "title": "three cubes 114 height 1e20",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}