{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R562",
  "content_hash": null,
  "slug": "pcpsg-attempt-literature-and-remaining-range",
  "type": "attempt",
  "title": "The final nine tenths of the requested base interval remain unswept",
  "summary": "The full answer lies between 2 and 49,600; this entry settles the prefix through 100 billion.",
  "relevance": "For Closest prime square to the cube of a prime below one trillion, record pcpsg-attempt-literature-and-remaining-range (“The final nine tenths of the requested base interval remain unswept”) documents a concrete method, search boundary, or failed route. The record states: The full answer lies between 2 and 49,600; this entry settles the prefix through 100 billion.",
  "relevance_source": "recorded",
  "body": "For the complete candidate interval, both \\(p\\) and each adjacent prime \\(q\\) are odd. Their cube and square are odd, so every positive difference is an even integer. The incumbent therefore gives the rigorous global enclosure\n\\[\n2\\leq\\min\\leq49600.\n\\]\nThe exhaustive computation in this entry proves equality with the upper endpoint when \\(p\\leq10^{11}\\). It makes no claim for\n\\[\n10^{11}<p\\leq10^{12}.\n\\]\n\nHall's 1971 paper introduced the general search for small nonzero \\(|x^3-y^2|\\). Elkies developed a lattice-reduction method for rational points near curves, and Aanderaa, Kristiansen, and Ruud later gave an algorithm that detects every Hall-good example inside its stated search space. Those works allow arbitrary integer square and cube bases. The present problem imposes primality on both bases and asks for an exact finite minimum.\n\nBaker, Harman, and Pintz prove that sufficiently large \\(x\\) has a prime in an interval of length \\(x^{0.525}\\). This gives asymptotic control of adjacent primes. Its scale is far larger than the interval in \\(q\\) corresponding to a square gap of 49,600 near \\(p^{3/2}\\), and the theorem does not identify the exact finite minimum here.\n\nFocused searches for the pair \\((1587809,2000771023)\\), its exact cube, prime-restricted Hall problems, and nearest prime squares found no primary source reporting this candidate. The status and novelty of the full trillion-base problem remain unverified.",
  "status": "partial",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the literature audit and the unresolved base interval 100000000000 < p <= 1000000000000",
    "bounds": {
      "p": {
        "min": 100000000001,
        "max": 1000000000000
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/math/0005139",
      "locator": "Marshall Hall, The Diophantine equation x^3-y^2=k, Computers in Number Theory (1971), 173-198; Noam Elkies, Rational points near curves and small nonzero |x^3-y^2| via lattice reduction, arXiv:math/0005139; Stål Aanderaa, Lars Kristiansen, and Hans Kristian Ruud, Search for good examples of Hall's conjecture, Mathematics of Computation 87 (2018), DOI 10.1090/MCOM/3298; R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83 (2001), DOI 10.1112/plms/83.3.532"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0005139",
    "locator": "Marshall Hall, The Diophantine equation x^3-y^2=k, Computers in Number Theory (1971), 173-198; Noam Elkies, Rational points near curves and small nonzero |x^3-y^2| via lattice reduction, arXiv:math/0005139; Stål Aanderaa, Lars Kristiansen, and Hans Kristian Ruud, Search for good examples of Hall's conjecture, Mathematics of Computation 87 (2018), DOI 10.1090/MCOM/3298; R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83 (2001), DOI 10.1112/plms/83.3.532"
  },
  "relations": [
    {
      "slug": "R565",
      "title": "The minimum gap through p=100 billion is 49,600",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "prime-cube-prime-square-gap-trillion",
      "title": "prime cube prime square gap trillion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}