{ "artifactId": "artifact.library.mathlib.pell-equation-solvability.v001", "candidateOnly": false, "category": "Number Theory", "claimBoundary": "This page indexes Mathlib's exact solvability equivalence for Pell's equation. With d : ℤ and 0 < d, it states that an integer pair (x, y) with x ^ 2 - d * y ^ 2 = 1 and y ≠ 0 exists if and only if ¬IsSquare d. The positivity hypothesis and the nontriviality condition y ≠ 0 are essential parts of the selected declaration. It does not classify solutions, assert infinitude, identify a fundamental or least solution, supply an algorithm, or cover nonpositive d.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Pell.exists_iff_not_isSquare {d : ℤ} (h₀ : 0 < d) : (∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0) ↔ ¬IsSquare d", "family": "Diophantine equations", "id": "library.mathlib.pell-equation-solvability.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/pell-equation-solvability/", "entryData": "/data/library-theorems/pell-equation-solvability.json", "evidence": "/proofs/artifact.library.mathlib.pell-equation-solvability.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.pell-equation-solvability.v001.evidence.json", "source": "/sources/upstream/pell-equation-solvability/" }, "landmark": { "completeFormalType": "theorem Pell.exists_iff_not_isSquare {d : ℤ} (h₀ : 0 < d) : (∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0) ↔ ¬IsSquare d", "concepts": [ "Pell equation", "Diophantine equations", "integer squares", "irrational square roots", "Diophantine approximation", "modular pigeonhole argument" ], "plainLanguage": "For a positive integer d, the Pell equation has an integer solution with y nonzero exactly when d is not a square.", "poster": { "alt": "An ivory number-theory poster places the positive integer parameter d above the exact Pell equation equivalence, then traces irrational square-root approximation, repetition of a bounded Pell-form value, a modular collision, and the square-case factorization obstruction.", "byteSize": 3061087, "caption": "For 0 < d, a nontrivial integer solution of x² − d y² = 1 exists exactly when d is not a square.", "description": "The poster states the exact iff for d : ℤ under 0 < d. Its checked-route section moves from irrationality of √d through rational approximation, a repeated nonzero value m and congruent numerator-denominator pairs, to a nontrivial Pell solution; the reverse direction factors the equation when d = a². The exact-scope footer excludes classification, infinitude, and minimal-solution claims.", "derivatives": [ { "byteSize": 128040, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-poster-v2-640.webp", "sha256": "sha256:3b56437f460a493ee7ce625a9fdafe47cb9eb32584e06f1f596bff0ba6263abd", "width": 640 }, { "byteSize": 265758, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-poster-v2-1024.webp", "sha256": "sha256:a18b973f5a8ed68944f6f6726b3bf9d913af0aeb9d6b1cc9af339463e745ba09", "width": 1024 } ], "generatedAt": "2026-07-27T22:36:55-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": null, "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-poster-v2.png", "sha256": "sha256:a0b23fa927d230674afedfe133be576fa2fb76ee165dff023d9ff62b74e51444", "title": "Pell equation solvability at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · DIOPHANTINE EQUATIONS\nPELL EQUATION SOLVABILITY\nd ∈ ℤ · 0 < d\n∃ x y ∈ ℤ, x² − d y² = 1 AND y ≠ 0\nIF AND ONLY IF d IS NOT A SQUARE\nTHE CHECKED ROUTE\n1 · A nonsquare d makes √d irrational\n2 · Rational approximations repeat one nonzero Pell value m\n3 · A modular collision produces a nontrivial solution\n4 · If d = a², factorization rules one out\nEXACT SCOPE\nOne existence equivalence. No classification, infinitude, or minimal solution." }, "visual": { "alt": "A dark engraved integer lattice carries the two branches of a Pell hyperbola. Four symmetry-related green off-axis lattice points sit on the nonsquare side, while a square-parameter inset factors into crossing diagonals and retains only the two y = 0 points; opposing gold arcs bind the two directions of the equivalence.", "byteSize": 2444242, "caption": "The nonsquare condition and the existence of an off-axis integer point on x² − d y² = 1 are the two directions of the exact positive-d equivalence.", "description": "The landscape separates the theorem's two logical directions without turning them into a classification. The main integer lattice highlights a symmetry-related set of nontrivial points with y ≠ 0 on a Pell hyperbola under 0 < d and ¬IsSquare d. A contrasting square-case factorization inset shows why d = a² leaves no off-axis integer solution. Opposing gold arcs connect the condition and existence motifs.", "derivatives": [ { "byteSize": 46042, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-schematic-v2-640.webp", "sha256": "sha256:5194dcc59502de3313f4e66e53766df24bbd2667862fbb20e368b896fd5f66d3", "width": 640 }, { "byteSize": 122126, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-schematic-v2-1200.webp", "sha256": "sha256:5fd33068ef621734044f3a71182768e95a5e4e1dde42cbc710d221f7ba8086e0", "width": 1200 } ], "generatedAt": "2026-07-27T22:11:52-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/pell-equation-solvability/theorem-schematic-v2.png", "sha256": "sha256:92cb61c652ffbb57aa39bd8c7745c7b641ea858d7efb12b6926610a9767a535f", "title": "A nontrivial lattice point exactly for nonsquare d", "width": 1536 }, "walkthrough": null, "whyLandmark": "Pell's equation is a central Diophantine equation, and this declaration captures its basic solvability boundary as an exact equivalence. Mathlib's source is especially instructive because it constructs the nonsquare direction from irrational approximation and a finite congruence collision, then proves the converse by factorization when d is a square." }, "nonClaims": [ "The theorem assumes 0 < d; it is not a statement for every integer d.", "The conclusion is the existence of at least one integer solution with y ≠ 0, not a classification or enumeration of all solutions.", "The declaration does not prove infinitude of the solution set or identify a fundamental, unique, canonical, or minimal solution.", "The checked proof uses irrational approximation and a finite congruence collision; Mathlib's source explicitly leaves a connection to continued fractions as future work.", "ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.pell-equation-solvability.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": "For positive d, the Pell equation has a nontrivial integer solution if and only if d is not a square.", "targetId": "target.library.mathlib.pell-equation-solvability.v001", "title": "Pell Equation Solvability", "upstreamOrigin": { "declarationName": "Pell.exists_iff_not_isSquare", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/Pell.lean#L388", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:ac09716ec72e9d69cc505b7505473a1734408c36178f361155543952aafe86cb", "sourceByteLength": 29404, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/Pell.lean", "sourceLine": 388, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }