{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R564",
  "content_hash": null,
  "slug": "pcpsg-claim-nearest-integer-reduction",
  "type": "claim",
  "title": "Two nearest integer squares suffice for any proposed gap bound",
  "summary": "Every square closest to p^3 has base floor(sqrt(p^3)) or its successor.",
  "relevance": "For Closest prime square to the cube of a prime below one trillion, record pcpsg-claim-nearest-integer-reduction (“Two nearest integer squares suffice for any proposed gap bound”) records a bound, answer, status fact, or structural consequence. The record states: Every square closest to p^3 has base floor(sqrt(p^3)) or its successor.",
  "relevance_source": "recorded",
  "body": "Put\n\\[\nr=\\lfloor\\sqrt{p^3}\\rfloor.\n\\]\nThe squares increase strictly with their nonnegative integer bases. Hence every \\(q\\leq r\\) satisfies \\(q^2\\leq r^2\\), while every \\(q\\geq r+1\\) satisfies \\(q^2\\geq(r+1)^2\\). It follows that\n\\[\n\\min_{q\\in\\mathbf Z_{\\geq0}}|p^3-q^2|\n=\\min\\{p^3-r^2,(r+1)^2-p^3\\}.\n\\]\n\nConsequently, a prime square whose gap is at most 49,600 must first appear among these same two integer squares. The sweep can rule out a better prime square by classifying only the nearest-integer candidates whose raw gaps meet that threshold. Any adjacent prime farther away has a larger square gap on its side.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "every positive integer p and every square base q when comparing q^2 with p^3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1401.4345",
      "locator": "Elementary monotonicity argument used by the exact sweep; Hall-type nearest-square search context in Aanderaa, Kristiansen, and Ruud"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1401.4345",
    "locator": "Elementary monotonicity argument used by the exact sweep; Hall-type nearest-square search context in Aanderaa, Kristiansen, and Ruud"
  },
  "relations": [
    {
      "slug": "R565",
      "title": "The minimum gap through p=100 billion is 49,600",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "prime-cube-prime-square-gap-trillion",
      "title": "prime cube prime square gap trillion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}