{
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  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList",
  "family": "Prime factorization",
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    "completeFormalType": "theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList",
    "concepts": [
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      "unique factorization",
      "list permutation",
      "natural numbers"
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    "plainLanguage": "Prime factors are unique apart from their order: if a list contains only primes and multiplies to n, it contains exactly the same prime factors, with the same multiplicities, as n.primeFactorsList.",
    "poster": {
      "alt": "Editorial theorem poster states the prime-list product and primality hypotheses, rearranges five distinct factor stones, and concludes uniqueness up to permutation with a four-step reading guide.",
      "byteSize": 2768658,
      "caption": "Mathlib's selected declaration gives the uniqueness of prime-factor lists up to permutation.",
      "description": "The retained poster presents the exact hypotheses, permutation conclusion, reading sequence, and boundary of Nat.primeFactorsList_unique without adding a factorization algorithm or ordered-list claim.",
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      "generatedAt": "2026-07-22T05:15:00Z",
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      "sha256": "sha256:060fbba25df9890710ceeca98c0407ebf715ec810dc2d9728a3736096073580c",
      "title": "Fundamental Theorem of Arithmetic — exact Mathlib form",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "NUMBER THEORY · PRIME-FACTOR UNIQUENESS\nFUNDAMENTAL THEOREM OF ARITHMETIC\nl.prod = n\nEvery member of l is prime\nl.Perm n.primeFactorsList\nSAME PRODUCT\nPRIME ENTRIES\nUNIQUE UP TO PERMUTATION\nHOW TO READ IT\n1 · Start with a finite prime list l\n2 · Multiply its entries to get n\n3 · Compare with n.primeFactorsList\n4 · The lists differ only by order\nEXACT SCOPE\nUniqueness up to permutation. No factorization algorithm or ordered-list claim."
    },
    "visual": {
      "alt": "Five distinct engraved stones follow gold permutation paths to the same five stones rearranged below.",
      "byteSize": 2953066,
      "caption": "A prime list with product n has the same entries, including multiplicity, as n.primeFactorsList; only their order may differ.",
      "description": "The theorem compares any finite all-prime list with the canonical prime-factor list after establishing that both multiply to the same natural number.",
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      "generatedAt": "2026-07-21T22:52:14Z",
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      "sha256": "sha256:185d73495447b0e808baf3620cb43c38c7a300cde7e65ef907dcd084d043cb6b",
      "title": "The same prime factors, in another order",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Unique prime factorization is the organizing principle behind elementary arithmetic and much of number theory. This declaration isolates its uniqueness half in an exact finite-list form."
  },
  "nonClaims": [
    "This selected declaration does not provide a factorization algorithm.",
    "It proves uniqueness up to permutation, not equality of arbitrarily ordered lists.",
    "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.fundamental-theorem-arithmetic.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": "Any finite list of primes whose product is n is a permutation of Mathlib's canonical prime-factor list for n.",
  "targetId": "target.library.mathlib.fundamental-theorem-arithmetic.v001",
  "title": "Fundamental Theorem of Arithmetic",
  "upstreamOrigin": {
    "declarationName": "Nat.primeFactorsList_unique",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Data/Nat/Factors.lean#L168",
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}