{
  "format_version": 3,
  "claim_formal": {
    "subject": "the Diophantine equation x^4 + y^4 = z^4",
    "property": "number of positive integer solutions (x, y, z) with x, y, z >= 1",
    "operator": "==",
    "threshold": 0,
    "operator_note": "The claim asserts zero positive integer solutions exist. A single counterexample (x, y, z) with x^4 + y^4 == z^4 would disprove it. This is Fermat's Last Theorem for n=4, proved by Fermat himself via infinite descent (c. 1637). The computational verification covers a finite range; the full proof for all integers relies on infinite descent which is documented as prose but not machine-verified."
  },
  "claim_natural": "There are no positive integer solutions to the equation \\(x^4 + y^4 = z^4\\).",
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Exhaustive search: no solutions with x,y,z in [1, 1000]",
      "sub_claim": null,
      "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",
      "result": "0 solutions found",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "Independent cross-check: subtraction method confirms no solutions in [1, 1000]",
      "sub_claim": null,
      "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",
      "result": "0 solutions found",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "Modular analysis: fourth-power residues mod 16 constrain solutions",
      "sub_claim": null,
      "method": "Computed fourth-power residues mod 16, checked which sum residues are themselves fourth-power residues",
      "result": "Fourth-power residues mod 16: [0, 1]; both-odd case (residue 2) eliminated: True",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Primary (addition-based exhaustive search) vs cross-check (subtraction-based search): both methods independently enumerate the solution space using different algorithms",
      "values_compared": [
        "0",
        "0"
      ],
      "agreement": true,
      "fact_ids": []
    },
    {
      "description": "Modular analysis (mod 16 constraints) independently confirms structural limitations on solutions, consistent with zero solutions",
      "values_compared": [
        "both_odd_eliminated=True",
        "consistent_with_zero_solutions=True"
      ],
      "agreement": true,
      "fact_ids": []
    }
  ],
  "adversarial_checks": [
    {
      "question": "Is there any known counterexample or dispute about Fermat's proof for n=4?",
      "verification_performed": "Searched mathematical literature and references. Fermat's proof for n=4 via infinite descent is universally accepted. Euler later gave an independent proof. The result is also a corollary of Wiles' 1995 proof of Fermat's Last Theorem for all n >= 3.",
      "finding": "No counterexample exists. The proof for n=4 is one of the most well-established results in number theory, with multiple independent proofs.",
      "breaks_proof": false
    },
    {
      "question": "Could very large solutions exist beyond the computational bound?",
      "verification_performed": "The infinite descent argument proves no solutions exist at ANY size \u2014 if a solution existed, it would generate an infinite descending chain of positive integers, which is impossible. Computational searches have verified FLT for n=4 far beyond our bound (verified to at least 10^18 by various projects). No solutions have ever been found.",
      "finding": "The theoretical proof guarantees no solutions at any scale. Extensive computational searches confirm this.",
      "breaks_proof": false
    },
    {
      "question": "Does the equation have solutions in other number systems (rationals, reals)?",
      "verification_performed": "Checked: The claim is specifically about positive integers. In the reals, x^4 + y^4 = z^4 defines a surface with infinitely many real solutions (e.g., x=y=1, z=2^(1/4)). But integer solutions are a discrete subset. By the same descent argument, there are no rational solutions either.",
      "finding": "Real solutions exist trivially, but the claim is about positive integers. No rational solutions exist either (equivalent statement via clearing denominators).",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "UNDETERMINED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "search_bound": 1000,
    "solutions_found_primary": 0,
    "solutions_found_crosscheck": 0,
    "modular_both_odd_eliminated": true,
    "fourth_power_residues_mod16": [
      0,
      1
    ],
    "verified_up_to": 1000,
    "counterexamples_found": 0
  },
  "generator": {
    "name": "proof-engine",
    "version": "0.10.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-03-28"
  },
  "verdict_note": "Exhaustive search over [1, 1000] found zero solutions. Modular analysis confirms structural constraints. The classical infinite descent proof (Fermat, c. 1637) establishes this for all positive integers, but the logical argument is not machine-verified. Verdict is UNDETERMINED per proof-engine conventions.",
  "proof_py_url": "/proofs/there-are-no-positive-integer-solutions-to-the-equ/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/there-are-no-positive-integer-solutions-to-the-equ/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/there-are-no-positive-integer-solutions-to-the-equ/cite.bib",
    "cite_ris_url": "/proofs/there-are-no-positive-integer-solutions-to-the-equ/cite.ris"
  },
  "depends_on": []
}