{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R485",
  "content_hash": null,
  "slug": "m127ac-claim-brauer-upper",
  "type": "claim",
  "title": "An explicit addition chain reaches the target in 136 steps",
  "summary": "Brauer's star-chain construction expands a ten-step chain for 127 into a checked 136-step chain for 2^127-1.",
  "relevance": "For Shortest addition chain for the 127th Mersenne number, record m127ac-claim-brauer-upper (“An explicit addition chain reaches the target in 136 steps”) records a bound, answer, status fact, or structural consequence. The record states: Brauer's star-chain construction expands a ten-step chain for 127 into a checked 136-step chain for 2^127-1.",
  "relevance_source": "recorded",
  "body": "For \\(M_r=2^r-1\\), use the star chain\n\\[\n1,2,3,5,7,14,28,56,63,126,127.\n\\]\nEach new exponent has the form \\(a+b\\), where \\(a\\) is the preceding exponent and \\(b\\) already occurs. Starting with \\(M_a\\), perform \\(b\\) doublings and then add the earlier value \\(M_b\\). The identity\n\\[\n2^bM_a+M_b=M_{a+b}\n\\]\nshows that this reaches the next Mersenne value.\n\nThe exponent increments sum to \\(127-1=126\\). There is one final addition for each of the ten star steps, so the expanded chain has length\n\\[\n126+10=136.\n\\]\nThe complete 137-value chain is stored in `m127ac-artifact-chain-replay`. Its final three values are\n\\[\n2^{126}-2,\\quad 2^{127}-2,\\quad 2^{127}-1.\n\\]\nThe artifact checks every predecessor pair and the final target.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the target N=2^127-1 and the exponent star chain (1,2,3,5,7,14,28,56,63,126,127)",
    "bounds": {
      "chain_length": {
        "min": 136,
        "max": 136
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1090/S0002-9904-1939-07068-7",
      "locator": "Alfred Brauer, On addition chains, Bulletin of the American Mathematical Society 45(10), 1939, pages 736-739; explicit specialization and replay in this dataset"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1090/S0002-9904-1939-07068-7",
    "locator": "Alfred Brauer, On addition chains, Bulletin of the American Mathematical Society 45(10), 1939, pages 736-739; explicit specialization and replay in this dataset"
  },
  "relations": [
    {
      "slug": "R483",
      "title": "Replayable 136-step addition chain",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R486",
      "title": "The strongest located interval is 133 through 136",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "mersenne-127-addition-chain",
      "title": "mersenne 127 addition chain",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}