{ "artifactId": "artifact.library.mathlib.fermat-last-theorem-exponent-four.v001", "candidateOnly": false, "category": "Number theory", "claimBoundary": "This target indexes Mathlib's theorem FermatLastTheoremFor 4. Unfolded, for natural numbers a, b, and c, if all three are nonzero then a ^ 4 + b ^ 4 ≠ c ^ 4. The source proves this through a stronger integer square equation and infinite descent. The declaration is the exponent-four case only, not full Fermat's Last Theorem, an exponent-general theorem, or new ProofAtlas mathematics.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem fermatLastTheoremFour : FermatLastTheoremFor 4", "family": "Diophantine equations", "id": "library.mathlib.fermat-last-theorem-exponent-four.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/fermat-last-theorem-exponent-four/", "entryData": "/data/library-theorems/fermat-last-theorem-exponent-four.json", "evidence": "/proofs/artifact.library.mathlib.fermat-last-theorem-exponent-four.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.fermat-last-theorem-exponent-four.v001.evidence.json", "source": "/sources/upstream/fermat-last-theorem-exponent-four/" }, "landmark": { "completeFormalType": "theorem fermatLastTheoremFour : FermatLastTheoremFor 4", "concepts": [ "Fermat's Last Theorem", "exponent four", "Diophantine equations", "infinite descent", "minimal counterexamples", "primitive Pythagorean triples" ], "plainLanguage": "No three nonzero natural numbers a, b, and c satisfy a⁴ + b⁴ = c⁴.", "poster": { "alt": "An ivory number-theory poster states the exponent-four impossibility for nonzero natural numbers, then shows a large square construction passing through two geometric decompositions to a smaller copy and lists the five-step descent.", "byteSize": 2326763, "caption": "A hypothetical exponent-four solution enters a stronger integer square equation and descends to a smaller solution, contradicting minimality.", "description": "The poster centers the exact nonzero-natural conclusion a⁴+b⁴≠c⁴. Below it, an unlabeled engraved square construction shrinks through two successive right-triangle decompositions. A compact stronger-equation panel and five-step route explain the least-counterexample contradiction while the footer limits the result to exponent four.", "derivatives": [ { "byteSize": 99102, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-poster-v1-640.webp", "sha256": "sha256:861990cf0ef460188840ebb05c16c506b5b499b2e0998649640d94c55895b808", "width": 640 }, { "byteSize": 193882, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-poster-v1-1024.webp", "sha256": "sha256:ef41a26ff39aaa6d7b5fcb625d347ac4db45d68b9ade7968d95eb4cfe420a3e8", "width": 1024 } ], "generatedAt": "2026-07-25T16:31:17-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-poster-v1.png", "sha256": "sha256:02152d156340c67e5e28417a7ec9247a9d260ccac359cbe8049734f00e3f7295", "title": "Fermat's Last Theorem for Exponent Four — infinite descent", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · INFINITE DESCENT\nFERMAT'S LAST THEOREM\nFOR EXPONENT FOUR\na,b,c ∈ ℕ and a,b,c ≠ 0\na⁴ + b⁴ ≠ c⁴\nA STRONGER INTEGER IMPOSSIBILITY\na⁴ + b⁴ = c² has no solution with a,b ≠ 0\nHOW THE CHECKED ROUTE DESCENDS\n1 · Choose a solution with least |c|\n2 · Make c positive, a odd, and gcd(a,b)=1\n3 · Classify two primitive Pythagorean triples\n4 · Obtain j⁴+k⁴=i² with |i|<|c|\n5 · Contradict the least choice\nEXACT SCOPE\nExponent 4 only. Not the full Fermat's Last Theorem." }, "visual": { "alt": "On a dark forest-green field, two large square-within-square tiles meet a double right-triangle construction, while a gold curve carries the same configuration into a much smaller nested copy.", "byteSize": 3088545, "caption": "Two Pythagorean decompositions force a hypothetical solution of the stronger square equation to reproduce at a strictly smaller scale.", "description": "Two fourth-power tiles enter a large ivory square-and-right-angle construction. A cobalt intermediate records the nested Pythagorean geometry, and an antique-gold descent arc lands on a smaller homologous configuration inside a circled aperture, making the least-versus-smaller conflict visible without embedded text.", "derivatives": [ { "byteSize": 61138, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-schematic-v1-640.webp", "sha256": "sha256:c2034a8f4fd06fe4c2136a68d49d3c34eb41e82fa1d6f569e8fab953246d67e0", "width": 640 }, { "byteSize": 215958, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-schematic-v1-1200.webp", "sha256": "sha256:c9d5d70c45e6368652250a1d0237c021e37c15de8564d97f2d2967f427e58a71", "width": 1200 } ], "generatedAt": "2026-07-25T16:31:17-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/fermat-last-theorem-exponent-four/theorem-schematic-v1.png", "sha256": "sha256:f879cfbfed0d4cea8c773394e349050b16f33ce49623440e827215c078642147", "title": "A fourth-power counterexample descends below its own minimum", "width": 1536 }, "walkthrough": null, "whyLandmark": "The exponent-four case is a classical landmark in Diophantine analysis: a minimal hypothetical solution to a stronger square equation is transformed into a smaller one, giving a canonical example of infinite descent." }, "nonClaims": [ "The checked declaration proves the case n = 4 only; it does not prove Fermat's Last Theorem for every exponent.", "The exact statement is over natural numbers and assumes that a, b, and c are all nonzero.", "The stronger integer equation a ^ 4 + b ^ 4 = c ^ 2 is the source's proof route, not an expansion of the declaration's public statement.", "The theorem is an impossibility result; it does not enumerate solutions, provide a finite search bound, or assert a computational method.", "ProofAtlas is indexing an existing Mathlib theorem, not claiming mathematical novelty." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.fermat-last-theorem-exponent-four.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Nonzero natural numbers cannot satisfy a⁴ + b⁴ = c⁴.", "targetId": "target.library.mathlib.fermat-last-theorem-exponent-four.v001", "title": "Fermat's Last Theorem for Exponent Four", "upstreamOrigin": { "declarationName": "fermatLastTheoremFour", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/FLT/Four.lean#L266", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:d81287497ab13720be761ccbbbd9e3bda3634f14fd83223b61f902d251848220", "sourceByteLength": 11535, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/FLT/Four.lean", "sourceLine": 266, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }