{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R610",
  "content_hash": null,
  "slug": "ptm61-claim-excluded-through-2pow28",
  "type": "claim",
  "title": "No trinomial multiple occurs through degree 2^28",
  "summary": "Exact residue search excludes every trinomial multiple with \\(0<a<b\\le2^{28}\\); existence and the least pair in the remaining range \\(2^{28}<b\\le2^{31}\\) remain open.",
  "relevance": "For Least trinomial multiple of a primitive degree-61 polynomial, record ptm61-claim-excluded-through-2pow28 (“No trinomial multiple occurs through degree 2^28”) records a bound, answer, status fact, or structural consequence. The record states: Exact residue search excludes every trinomial multiple with \\(0<a<b\\le2^{28}\\); existence and the least pair in the remaining range \\(2^{28}<b\\le2^{31}\\) remain open.",
  "relevance_source": "recorded",
  "body": "Let \\(\\alpha\\) be the residue class of \\(x\\) in\n\\[\n\\mathbf F_2[x]/(x^{61}+x^{45}+x^{32}+x^2+1).\n\\]\nFor \\(0<a<b\\), divisibility by \\(f\\) is equivalent to\n\\[\n\\alpha^b+\\alpha^a+1=0,\n\\]\nor \\(\\alpha^a=\\alpha^b+1\\).\n\nThe certificate stores every tagged residue \\((\\alpha^a,a)\\) for\n\\[\n1\\leq a<2^{28}.\n\\]\nIt then tests \\(\\alpha^b+1\\) in increasing order for every \\(1\\leq b\\leq2^{28}\\), scans the complete bucket containing that residue, and accepts only a matching tag with \\(a<b\\). It inspected 536,886,339 packed records and found no match. Thus\n\\[\nf(x)\\nmid x^b+x^a+1\n\\]\nfor every \\(0<a<b\\leq268435456\\).\n\nThis proves one eighth of the requested degree interval. The 1,879,048,192 possible values of \\(b\\) satisfying \\(2^{28}<b\\leq2^{31}\\) remain outside the certified search.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every exponent pair 0 < a < b <= 268435456 for the stated degree-61 polynomial over F_2",
    "bounds": {
      "b": {
        "min": 1,
        "max": 268435456
      },
      "polynomial_degree": {
        "min": 61,
        "max": 61
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/cs/0701069",
      "locator": "Exact exhaustive computation in ptm61-artifact-bucket-exclusion-2pow28, executed on 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/cs/0701069",
    "locator": "Exact exhaustive computation in ptm61-artifact-bucket-exclusion-2pow28, executed on 2026-07-24"
  },
  "models": [],
  "relations": [
    {
      "slug": "R607",
      "title": "Exact 2^28 bucket exclusion certificate",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R611",
      "title": "The stated polynomial gives a primitive degree-61 field model",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R609",
      "title": "The interval above 2^28 remains open in this entry",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "primitive-degree61-trinomial-multiple",
      "title": "primitive degree61 trinomial multiple",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}