{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R767",
  "content_hash": null,
  "slug": "tc114-claim-height-1e20-open",
  "type": "claim",
  "title": "The height-10^20 question remains open",
  "summary": "No verified representation of 114 is known, and the published computations do not certify the full height box.",
  "relevance": "For A bounded three-cubes search for 114, record tc114-claim-height-1e20-open (“The height-10^20 question remains open”) records a bound, answer, status fact, or structural consequence. The record states: No verified representation of 114 is known, and the published computations do not certify the full height box.",
  "relevance_source": "recorded",
  "body": "Define the height by\n\\[\nH(x,y,z)=\\max\\{|x|,|y|,|z|\\}.\n\\]\nThe target asks for an integer triple satisfying\n\\[\nx^3+y^3+z^3=114,\\qquad H(x,y,z)\\leq10^{20}.\n\\]\nPermutations count as the same representation for search purposes, although the equation itself uses ordered variables.\n\nBooker and Sutherland listed 114 among the seven cases below 1,000 that remained unresolved in 2021. Their September 2019 computation included 114 and covered every solution with\n\\[\n\\min\\{|x|,|y|,|z|\\}\\leq10^{17}.\n\\]\nTheir later computation reached a larger rectangle in the search parameters, rather than the full height box. Grantham and Walsh's arXiv preprint, submitted on 22 November 2022 as version 1, reports that their substantial effort on \\(k=114\\) remained unsuccessful. Thus the current source record supplies neither a solution of height at most \\(10^{20}\\) nor a complete exclusion certificate.\n\nThe fully covered prefix leaves a precise unresolved part of the requested box:\n\\[\n10^{17}<\\min\\{|x|,|y|,|z|\\}\n\\leq H(x,y,z)\\leq10^{20}.\n\\]\nSome of this region lies inside the later parameter search recorded separately.",
  "status": "open",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "integer triples with max(abs(x), abs(y), abs(z)) at most 10^20",
    "bounds": {
      "height": {
        "min": 0,
        "max": 100000000000000000000
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2211.12149v1",
      "locator": "Jon Grantham and P. G. Walsh, Representing integers as a sum of three cubes, arXiv:2211.12149v1, submitted 22 November 2022, closing sentence of Section 2; the authors report that their significant effort on k=114 remained unsuccessful"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2211.12149v1",
    "locator": "Jon Grantham and P. G. Walsh, Representing integers as a sum of three cubes, arXiv:2211.12149v1, submitted 22 November 2022, closing sentence of Section 2; the authors report that their significant effort on k=114 remained unsuccessful"
  },
  "relations": [
    {
      "slug": "R768",
      "title": "The published complete-search frontier is 10^17 in the smallest coordinate",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R770",
      "title": "A later partial search reached |z|=10^19",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R769",
      "title": "Every coordinate is 2 modulo 3",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "three-cubes-114-height-1e20",
      "title": "three cubes 114 height 1e20",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}