{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R565",
  "content_hash": null,
  "slug": "pcpsg-claim-prefix-minimum-100-billion",
  "type": "claim",
  "title": "The minimum gap through p=100 billion is 49,600",
  "summary": "An exact sweep over all 4,117,976,315 prime bases through \\(p=10^{11}\\) finds prefix minimum 49,600 at \\(p=1587809\\) and \\(q=2000771023\\); the range \\(10^{11}<p\\le10^{12}\\) remains unswept, so the requested full-range minimum remains open.",
  "relevance": "For Closest prime square to the cube of a prime below one trillion, record pcpsg-claim-prefix-minimum-100-billion (“The minimum gap through p=100 billion is 49,600”) records a bound, answer, status fact, or structural consequence. The record states: An exact sweep over all 4,117,976,315 prime bases through \\(p=10^{11}\\) finds prefix minimum 49,600 at \\(p=1587809\\) and \\(q=2000771023\\); the range \\(10^{11}<p\\le10^{12}\\) remains unswept, so the requested full-range minimum remains open.",
  "relevance_source": "recorded",
  "body": "For every prime \\(p\\) satisfying\n\\[\n10^6\\leq p\\leq10^{11},\n\\]\nthe smaller distance between \\(p^3\\) and the square of either adjacent prime around \\(p^{3/2}\\) is at least 49,600. Equality occurs at\n\\[\np=1587809,\\qquad q_-(p)=2000771023,\n\\]\nwhere\n\\[\np^3=4003084686476516129,\n\\]\n\\[\nq_-(p)^2=4003084686476466529,\n\\]\nand\n\\[\np^3-q_-(p)^2=49600.\n\\]\n\nThe segmented sieve enumerated 4,117,976,315 prime bases in the stated interval. For each base it computed the exact integer square root of \\(p^3\\). Only four nearest-integer square gaps were at most 49,600. The displayed pair supplies the sole prime square among them. The other three square bases are even:\n\\[\n2113144738,\\quad59120053422,\\quad1049747744368.\n\\]\nTheir gaps are 24,315, 17,767, and 25,895. Since their square bases are composite, they cannot enter the prime-restricted minimum.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every prime base p satisfying 1000000 <= p <= 100000000000",
    "bounds": {
      "p": {
        "min": 1000000,
        "max": 100000000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/0005139",
      "locator": "Exact sweep in pcpsg-artifact-segmented-sweep-100-billion and primality certificate in pcpsg-artifact-pocklington-incumbent"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0005139",
    "locator": "Exact sweep in pcpsg-artifact-segmented-sweep-100-billion and primality certificate in pcpsg-artifact-pocklington-incumbent"
  },
  "relations": [
    {
      "slug": "R561",
      "title": "Segmented prime sweep through 100 billion",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R564",
      "title": "Two nearest integer squares suffice for any proposed gap bound",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R563",
      "title": "Pocklington certificates prove both numbers in the incumbent pair prime",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R562",
      "title": "The final nine tenths of the requested base interval remain unswept",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "prime-cube-prime-square-gap-trillion",
      "title": "prime cube prime square gap trillion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}