{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R411",
  "content_hash": null,
  "slug": "ipte11-claim-currently-open",
  "type": "claim",
  "title": "Integral ideal solutions of size eleven remain unknown",
  "summary": "The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.",
  "relevance": "For An ideal Prouhet-Tarry-Escott solution of size eleven, record ipte11-claim-currently-open (“Integral ideal solutions of size eleven remain unknown”) records a bound, answer, status fact, or structural consequence. The record states: The candidate supplies no size-eleven witness, and the strongest published search excludes primitive symmetric solutions only through height 3500.",
  "relevance_source": "recorded",
  "body": "No pair of eleven-element integer sets is supplied in the candidate record. Its displayed pair has twelve elements on each side and serves as a neighboring known case.\n\nCoppersmith, Mossinghoff, Scheinerman, and VanderKam state that integral ideal PTE solutions are known for sizes at most ten and for size twelve. Their 2024 paper reports no new integral solution at sizes nine through sixteen. For odd size eleven, symmetry means \\(B=-A\\). Section 4.1 and Table 2 give an exhaustive search for primitive symmetric ideal solutions of height at most \\(3500\\), where height is the largest absolute coordinate. The table records zero solutions. Their search extends the height-2000 computation of Borwein, Lisoněk, and Percival.\n\nThis is a bounded result inside the symmetric family. It leaves asymmetric solutions and symmetric solutions of greater height untouched. Common translation and nonzero scaling preserve the equations, so a future witness should state its affine normalization. For a symmetric witness, a centered representative is automatic; dividing by the common gcd gives a primitive representative.\n\nThe literature search checked the 2003 search paper, the 2024 Mathematics of Computation paper and its tables, and later PTE work available through July 24, 2026. It found no claimed integer witness of size eleven and no proof of general nonexistence. The mathematical status is open.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the published existence status of an integral ideal Prouhet-Tarry-Escott solution of size 11, checked on 2026-07-24",
    "bounds": {
      "size": {
        "min": 11,
        "max": 11
      },
      "degree": {
        "min": 10,
        "max": 10
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1090/mcom/3917",
      "locator": "Don Coppersmith, Michael J. Mossinghoff, Danny Scheinerman, and Jeffrey M. VanderKam, Ideal solutions in the Prouhet-Tarry-Escott problem, Mathematics of Computation 93 (2024), 2473-2501: Introduction, pages 2474-2475; Section 4.1 size n=11 result, page 2486; Table 2, page 2487. Peter Borwein, Petr Lisoněk, and Colin Percival, Computational investigations of the Prouhet-Tarry-Escott problem, Mathematics of Computation 72 (2003), 2063-2070: Table 2 and size-11 discussion, page 2069"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1090/mcom/3917",
    "locator": "Don Coppersmith, Michael J. Mossinghoff, Danny Scheinerman, and Jeffrey M. VanderKam, Ideal solutions in the Prouhet-Tarry-Escott problem, Mathematics of Computation 93 (2024), 2473-2501: Introduction, pages 2474-2475; Section 4.1 size n=11 result, page 2486; Table 2, page 2487. Peter Borwein, Petr Lisoněk, and Colin Percival, Computational investigations of the Prouhet-Tarry-Escott problem, Mathematics of Computation 72 (2003), 2063-2070: Table 2 and size-11 discussion, page 2069"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R410",
      "title": "The supplied size-twelve control witness is valid and primitive",
      "object_type": "artifact",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "ideal-pte-size-11",
      "title": "ideal pte size 11",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}