{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R152",
  "content_hash": null,
  "slug": "cmi-artifact-prime-sweep-50000",
  "type": "artifact",
  "title": "Exact image enumeration for 5,131 primes",
  "summary": "Direct enumeration confirms the formula for every prime below 50,000.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Standard-library Python sieves every prime below 50,000, evaluates \\(x^3+x\\) at every field element, and counts the marked residues. All 5,131 primes from 5 through 49,999 agree with \\(\\lfloor(2p+1)/3\\rfloor\\). The compact JSON vector of pairs \\([p,|V_p|]\\) has SHA-256 digest `68924e72573a23ff4bf1dff6597f0b27b6425fbd2d833b50f1f8f74a475b25a8`.\n\nSelected pairs are \\((5,3)\\), \\((7,5)\\), \\((11,7)\\), \\((13,9)\\), \\((101,67)\\), \\((1009,673)\\), \\((10007,6671)\\), and \\((49999,33333)\\). This extends the candidate's reported sweep below 10,000. The proof establishes every prime.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "every prime p from 5 through 49999",
    "bounds": {
      "p": {
        "min": 5,
        "max": 49999
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3, standard library only",
    "citation": {
      "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
      "locator": "Inline Python source below, independently executed on 2026-07-24"
    },
    "inline_source": [
      "from hashlib import sha256",
      "from json import dumps",
      "LIMIT=50000",
      "prime=bytearray([1])*LIMIT",
      "prime[0]=prime[1]=0",
      "for q in range(2,int((LIMIT-1)**0.5)+1):",
      "    if prime[q]:",
      "        count=(LIMIT-1-q*q)//q+1",
      "        prime[q*q:LIMIT:q]=bytearray(count)",
      "out=[]",
      "for p in range(5,LIMIT):",
      "    if not prime[p]: continue",
      "    seen=bytearray(p)",
      "    for x in range(p): seen[(x*x*x+x)%p]=1",
      "    size=sum(seen)",
      "    assert size==(2*p+1)//3",
      "    out.append([p,size])",
      "raw=dumps(out,separators=(',',':'))",
      "digest=sha256(raw.encode()).hexdigest()",
      "assert len(out)==5131 and out[0]==[5,3] and out[-1]==[49999,33333]",
      "assert digest=='68924e72573a23ff4bf1dff6597f0b27b6425fbd2d833b50f1f8f74a475b25a8'",
      "print(len(out),out[0],out[-1],digest)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.jnt.2024.04.016",
    "locator": "Inline Python source below, independently executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R154",
      "title": "The image of x cubed plus x has the proposed size",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "R155",
      "title": "The complete fiber profile follows from one character sum",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "cubic-map-image-size",
      "title": "cubic map image size",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}