{ "artifactId": "artifact.library.mathlib.divergence-prime-reciprocal-series.v001", "candidateOnly": false, "category": "Number theory", "claimBoundary": "This target indexes Mathlib's order-independent subtype form: the real-valued family p ↦ 1/p on Nat.Primes is not Summable. The declaration is not literally phrased as convergence of ordered partial sums to +∞, and it supplies no divergence rate or asymptotic estimate.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ))", "family": "Prime reciprocal series", "id": "library.mathlib.divergence-prime-reciprocal-series.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/divergence-prime-reciprocal-series/", "entryData": "/data/library-theorems/divergence-prime-reciprocal-series.json", "evidence": "/proofs/artifact.library.mathlib.divergence-prime-reciprocal-series.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.divergence-prime-reciprocal-series.v001.evidence.json", "source": "/sources/upstream/divergence-prime-reciprocal-series/" }, "landmark": { "completeFormalType": "theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ))", "concepts": [ "reciprocals of primes", "nonsummable families", "smooth numbers", "rough numbers", "vanishing tails" ], "plainLanguage": "The real family that assigns the reciprocal 1/p to every prime p cannot have a finite sum.", "poster": { "alt": "An ivory number-theory poster states nonsummability over Nat.Primes, then shows smooth-and-rough counting feeding a prime-tail basin and a second remote cutoff with another retained basin beyond it.", "byteSize": 3099251, "caption": "Smooth and rough number counts force a finite reciprocal-prime tail of at least one half beyond every cutoff, contradicting tail vanishing under Summable.", "description": "The exact subtype-indexed conclusion leads into the checked Erdős route: split bounded naturals into smooth and rough parts, control the smooth part by a square-root estimate, charge the rough part to primes beyond one cutoff, and repeat the visible tail-basin mechanism beyond a farther cutoff. The second gate makes the obstruction to summable tails explicit rather than presenting one isolated finite sum.", "derivatives": [ { "byteSize": 164924, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-poster-v2-640.webp", "sha256": "sha256:e21419084ab85f69a1dd24d83665db754b9ac49c0f43577bfc4942413f5bcced", "width": 640 }, { "byteSize": 374742, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-poster-v2-1024.webp", "sha256": "sha256:3b8d61c5d211e8bcf928aa15938e0fafd57729e8954c0aa0d5bb05edfea1d1eb", "width": 1024 } ], "generatedAt": "2026-07-25T20:37:59Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-poster-v2.png", "sha256": "sha256:78cd16df93f5e3b22bdc8142d05f255e9d1f7a3c0f5d88bdf9f8c21dd5cdde9c", "title": "Divergence of Prime Reciprocals — a nonvanishing tail", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · PRIME RECIPROCALS\nDIVERGENCE OF PRIME RECIPROCALS\nTHE FAMILY IS NOT SUMMABLE\nIndex: p in Nat.Primes · Value: 1/p in the real numbers\nTAILS CANNOT VANISH\nTHE CHECKED ROUTE\n1 · Count smooth and rough numbers\n2 · Bound rough numbers by a finite prime tail\n3 · Force a tail sum of at least 1/2\n4 · Contradict vanishing tails under Summable\nEXACT SCOPE\nNonsummability over the subtype Nat.Primes.\nNot literally an ordered partial-sum limit to positive infinity." }, "visual": { "alt": "A dark engraved counting field splits natural-number tokens into a small smooth-number chamber and a rough-number stream supported by gold prime facets, forcing a fixed-height reciprocal-tail reservoir beyond every cutoff.", "byteSize": 3048369, "caption": "Beyond every cutoff, a finite block of primes still carries a fixed positive reciprocal mass, contradicting the vanishing tails required by summability.", "description": "The source counts integers by separating smooth numbers from rough numbers. The smooth portion fits beneath a square-root-sized vault, while each rough number is charged to a prime beyond the cutoff. The resulting lower bound fills a gold prime-tail reservoir to the same positive threshold no matter how far the cutoff moves.", "derivatives": [ { "byteSize": 51512, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-schematic-v1-640.webp", "sha256": "sha256:e9ffc1aae22c5aa937a6963598e63da7bf346e34b9e755342d2920c37d91d343", "width": 640 }, { "byteSize": 192184, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-schematic-v1-1200.webp", "sha256": "sha256:3b001a64450d942283b1a187d3a38ff34a9a9271252615e2ec9aa15f179a234a", "width": 1200 } ], "generatedAt": "2026-07-25T20:37:59Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/divergence-prime-reciprocal-series/theorem-schematic-v1.png", "sha256": "sha256:dca5ea430bc4ca6cb82a73c8134bd25aa2b664575d4cb3b4659540c735b90099", "title": "A fixed positive reciprocal-prime mass survives beyond every cutoff", "width": 1536 }, "walkthrough": null, "whyLandmark": "Euler's theorem that the sum of prime reciprocals diverges is a classical bridge between elementary series and the distribution of primes. Mathlib retains a particularly visual Erdős proof in which smooth and rough number counts force a uniform lower bound on arbitrarily remote finite prime tails." }, "nonClaims": [ "The exact endpoint is nonsummability over the subtype Nat.Primes, independent of any chosen enumeration.", "The declaration is not literally a statement that one ordered sequence of partial sums tends to +∞.", "It gives no quantitative divergence rate, asymptotic formula, or estimate for the prime-counting function.", "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.divergence-prime-reciprocal-series.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": "The reciprocal family indexed by the subtype of natural primes is not summable.", "targetId": "target.library.mathlib.divergence-prime-reciprocal-series.v001", "title": "Divergence of the Reciprocal-Prime Series", "upstreamOrigin": { "declarationName": "Nat.Primes.not_summable_one_div", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/SumPrimeReciprocals.lean#L87", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:7fe6a032b1a639a87b78dc67a444954fa3ec89a9b125d66de6047a0bdf042fe4", "sourceByteLength": 5115, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/SumPrimeReciprocals.lean", "sourceLine": 87, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }