{
  "prefix": {
    "pe": "https://proofengine.info/proofs/there-are-no-positive-integer-solutions-to-the-equ/",
    "prov": "http://www.w3.org/ns/prov#",
    "xsd": "http://www.w3.org/2001/XMLSchema#",
    "schema": "http://schema.org/"
  },
  "entity": {
    "pe:claim": {
      "prov:type": "pe:Claim",
      "prov:label": "There are no positive integer solutions to the equation \\(x^4 + y^4 = z^4\\).",
      "prov:value": "There are no positive integer solutions to the equation \\(x^4 + y^4 = z^4\\)."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Exhaustive search: no solutions with x,y,z in [1, 1000]",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Exhaustive search: iterated all (x, y) with 1 <= x <= y <= 1000, checked if x^4 + y^4 is a perfect fourth power using integer arithmetic",
      "pe:result": "0 solutions found"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Independent cross-check: subtraction method confirms no solutions in [1, 1000]",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Subtraction method: for each z in [2, 1000], for each x in [1, z-1], checked if z^4 - x^4 is a perfect fourth power",
      "pe:result": "0 solutions found"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Modular analysis: fourth-power residues mod 16 constrain solutions",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "Computed fourth-power residues mod 16, checked which sum residues are themselves fourth-power residues",
      "pe:result": "Fourth-power residues mod 16: [0, 1]; both-odd case (residue 2) eliminated: True"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: UNDETERMINED",
      "prov:value": "UNDETERMINED"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-03-28"
    }
  },
  "agent": {
    "pe:proof-engine": {
      "prov:type": "prov:SoftwareAgent",
      "prov:label": "Proof Engine",
      "schema:version": "0.10.0",
      "schema:url": "https://github.com/yaniv-golan/proof-engine"
    }
  },
  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
    }
  },
  "wasDerivedFrom": {},
  "wasAttributedTo": {
    "pe:attr-verdict": {
      "prov:entity": "pe:verdict",
      "prov:agent": "pe:proof-engine"
    }
  },
  "used": {
    "pe:used-A1": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A1"
    },
    "pe:used-A2": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A2"
    },
    "pe:used-A3": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A3"
    }
  }
}