{ "artifactId": "artifact.library.mathlib.lagrange-four-square-theorem.v001", "candidateOnly": false, "category": "Number theory", "claimBoundary": "This target indexes Mathlib's existence theorem that every natural number is a sum of four squares of natural numbers. It does not count representations, require all four terms to be nonzero, or claim a new proof.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Nat.sum_four_squares (n : ℕ) : ∃ a b c d : ℕ, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = n", "family": "Additive number theory", "id": "library.mathlib.lagrange-four-square-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/lagrange-four-square-theorem/", "entryData": "/data/library-theorems/lagrange-four-square-theorem.json", "evidence": "/proofs/artifact.library.mathlib.lagrange-four-square-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.lagrange-four-square-theorem.v001.evidence.json", "source": "/sources/upstream/lagrange-four-square-theorem/" }, "landmark": { "completeFormalType": "theorem Nat.sum_four_squares (n : ℕ) : ∃ a b c d : ℕ, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = n", "concepts": [ "sum of squares", "Euler four-square identity", "prime descent", "minimal multiplier", "multiplicative induction" ], "plainLanguage": "For every natural number n, there are four natural numbers—some of which may be zero—whose squares add exactly to n.", "poster": { "alt": "An ivory number-theory poster shows exactly four square tiles, Euler multiplication rosettes, a shrinking prime multiplier, the four-square equation, five proof movements, and existence-only scope.", "byteSize": 2689192, "caption": "Every natural number has a representation as four natural-number squares, with zero terms allowed.", "description": "The poster connects the exact four-square statement to Euler multiplication, the modular prime seed, least-multiplier descent, and final multiplicative induction without adding a representation count or uniqueness claim.", "derivatives": [ { "byteSize": 140184, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:fd4686c16c7dc2990df68d3a3eb59a41a06f8263fe2ab49657209ef3b301ae6d", "width": 640 }, { "byteSize": 290506, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:10c37e565212bea8302e466b70b368ea0a4805f0b23a8095efc351ab085e5de6", "width": 1024 } ], "generatedAt": "2026-07-22T01:26:27Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-poster-v1.png", "sha256": "sha256:e529768f32751cb0d978dcfcaedadcd0e6081920de997586cac89bbc03b630fd", "title": "Lagrange's Four-Square Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · FOUR SQUARES\nLAGRANGE'S FOUR-SQUARE THEOREM\nEvery natural number is a sum of four natural-number squares\nn = a² + b² + c² + d²\nFOUR-SQUARE REPRESENTATIONS MULTIPLY\nHOW THE CHECKED ROUTE MOVES\n1 · Euler's identity preserves four-square form under products\n2 · Seed a multiple of each prime with four squares\n3 · Choose the smallest possible positive multiplier\n4 · Descend to multiplier one\n5 · Induct over prime multiplication\nEXACT SCOPE\nExistence only. Zero terms are allowed; no representation count or uniqueness." }, "visual": { "alt": "Four-armed square-tipped rosettes combine through a gold Euler braid while nested multiplier collars descend around one prime token before final multiplicative assembly.", "byteSize": 3228808, "caption": "Euler multiplication and a least-multiplier prime descent assemble four-square representations for every natural number.", "description": "The recurring four-arm rosette represents one sum of four squares. Euler's braid preserves that form under multiplication, the multiplier around a prime shrinks to one, and prime representations then assemble into a representation of an arbitrary natural number.", "derivatives": [ { "byteSize": 63306, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:0be94e837fc05ffd2fe7fe699d4395b8ea91e24df83a6eaf4e734e5559a35b23", "width": 640 }, { "byteSize": 213864, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:c0ab7037c56dc70a3ea53a7dafa918d46acde194ef3ae20cf624b46ee454d694", "width": 1200 } ], "generatedAt": "2026-07-21T23:43:52Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/theorem-schematic-v1.png", "sha256": "sha256:816f20be43ce4e3e0ac2bed6b0138bd943a313da078a6ab674c88b4d238560b8", "title": "Lagrange's Four-Square Theorem schematic", "width": 1536 }, "walkthrough": { "introduction": "Mathlib proves that every natural number is a sum of four natural-number squares by establishing Euler's multiplicative identity, representing a multiple of each prime, descending a least positive multiplier to one, and extending prime representations to all naturals by multiplicative induction.", "pages": [ { "alt": "Two four-armed engraved rosettes with square ivory tips enter an antique-gold braid and emerge as one four-armed rosette.", "byteSize": 3260260, "caption": "Euler's identity combines two four-coordinate representations while preserving the sum-of-four-squares form of their product.", "description": "The proof first establishes Euler's four-square identity in a commutative ring and then a natural-number version using integer absolute values. This identity supplies the multiplicative step used both in prime descent and in the final induction.", "derivatives": [ { "byteSize": 44484, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-01-v1-640.webp", "sha256": "sha256:074b949d71f06d162097194b2ea09ee9d7051219b56ea3eb1c11be5c29972513", "width": 640 }, { "byteSize": 180714, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-01-v1-1200.webp", "sha256": "sha256:34a008bd346d692a9424a5bd4be6d2ddd267ad420c86813ffc2a8c66bc05454c", "width": 1200 } ], "generatedAt": "2026-07-21T23:28:42Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-01-v1.png", "sha256": "sha256:5bd6da9b6306fefb2fa007bb2e553490bc8d565f22c65cede38d874a1d145c1b", "title": "Two four-square representations multiply into another", "width": 1536, "equation": "(sum of four squares) · (sum of four squares) = sum of four squares", "kind": "proof_walkthrough_page", "sequenceIndex": 1 }, { "alt": "A four-armed representation rosette surrounds a faceted antique-gold prime token and an emerald multiplier collar prepared for descent.", "byteSize": 2885662, "caption": "A modular sum-of-two-squares argument yields a four-square representation of k p with the multiplier strictly between zero and p.", "description": "For an odd prime, Mathlib uses a congruence for a sum of two squares modulo p, adds the square of one and the square of zero, and extracts a positive multiplier smaller than the prime. The prime two is handled directly.", "derivatives": [ { "byteSize": 33936, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-02-v1-640.webp", "sha256": "sha256:160b4abce8d45196f6c623bfcbba1ea2cb0cd6bc0ad69f8877d50b0ed5df5936", "width": 640 }, { "byteSize": 129766, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-02-v1-1200.webp", "sha256": "sha256:c54fd3953edebfda57f79b5fc385e92a4dcbe5a3561769fd362f6732ac94f5cb", "width": 1200 } ], "generatedAt": "2026-07-22T01:26:27Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-02-v1.png", "sha256": "sha256:cef61e2b4c7714df3e8a93c3677a92d4aaad09b1dfb29df0b4c1fcb31b456a96", "title": "Start with four squares equal to a small positive multiple of a prime", "width": 1536, "equation": "0 < k < p and a²+b²+c²+d² = k p", "kind": "proof_walkthrough_page", "sequenceIndex": 2 }, { "alt": "Cobalt residue beads move within the four square tiles while nested emerald collars shrink concentrically around the same gold prime token.", "byteSize": 2922633, "caption": "Choose the least represented positive multiplier; halve it when even, or replace all four coordinates by least absolute residues when odd.", "description": "Nat.findX selects a minimal positive m with four squares equal to m p. If m is even, a parity lemma divides the representation by two. If m is odd, each coordinate is replaced by a congruent representative with absolute value less than half m, producing a positive quotient r1 impossible, so the prime itself is a sum of four squares.", "derivatives": [ { "byteSize": 32384, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-04-v1-640.webp", "sha256": "sha256:28416875ed95e33a18c2986e6f1d6fba51ebaec84cb339073cc23db89fe0edbb", "width": 640 }, { "byteSize": 119658, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-04-v1-1200.webp", "sha256": "sha256:f43e3b2f462fcc999f8bf29e9eebd15304cb024c92281cb647c254662de3f8da", "width": 1200 } ], "generatedAt": "2026-07-21T23:39:20Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-04-v1.png", "sha256": "sha256:4321cc423a5f711e6a2dbded6ab2011b19027e9d82fd842756b9906861fd10c1", "title": "The four-square identity completes the descent", "width": 1536, "equation": "m = 1; hence p = a²+b²+c²+d²", "kind": "proof_walkthrough_page", "sequenceIndex": 4 }, { "alt": "Several four-armed prime rosettes combine through recurring gold braids into one final complete four-square rosette, with blank ivory tiles allowed.", "byteSize": 3019359, "caption": "Zero and one are immediate, primes use the descent, and products use Euler's identity, completing multiplicative induction.", "description": "Nat.sum_four_squares invokes Nat.recOnMul. The zero and one cases use representations with zero terms; the prime case invokes the theorem from page 4; and the product case combines two inductive representations with Euler's identity. Therefore every natural number has a four-square representation.", "derivatives": [ { "byteSize": 42124, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-05-v1-640.webp", "sha256": "sha256:f258dbe31978015cec8485a745ab781eb0791e0bfadc97aa8ee89eb753a0d79b", "width": 640 }, { "byteSize": 152704, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-05-v1-1200.webp", "sha256": "sha256:5344604f9533057c31f7210902c70129a9fc18219660252a7e1ce1e5357228bd", "width": 1200 } ], "generatedAt": "2026-07-21T23:51:47Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/lagrange-four-square-theorem/proof-walkthrough-05-v1.png", "sha256": "sha256:1554c3157497cc74d831b5a09874a52dca8cfc11602a6a4ad0eb1440151643af", "title": "Multiplicative induction reaches every natural number", "width": 1536, "equation": "∀ n : ℕ, ∃ a b c d : ℕ, a²+b²+c²+d²=n", "kind": "proof_walkthrough_page", "sequenceIndex": 5 } ], "title": "Euler multiplication and Lagrange descent" }, "whyLandmark": "Lagrange's theorem gives a universal additive representation of every natural number. Mathlib's proof combines Euler's four-square identity, prime descent, and multiplicative induction." }, "nonClaims": [ "Proof Atlas did not originate Lagrange's Four-Square Theorem or Mathlib's declaration.", "The theorem asserts existence only: it neither counts representations nor requires every term to be nonzero.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.lagrange-four-square-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "2 of 3 presentation reviews recorded", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Every natural number is a sum of four squares of natural numbers.", "targetId": "target.library.mathlib.lagrange-four-square-theorem.v001", "title": "Lagrange's Four-Square Theorem", "upstreamOrigin": { "declarationName": "Nat.sum_four_squares", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/SumFourSquares.lean#L199", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:1bc309254aa0aabc39c9802eb6ec2358468f7b4d4cf13efdde8888995558eb4a", "sourceByteLength": 10569, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/SumFourSquares.lean", "sourceLine": 199, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }