Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Number theory · existing Mathlib theorem

Fundamental Theorem of Arithmetic

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.

Four separate status axes

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed page2 of 3 presentation reviews recorded
Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Formal orientation

Statement map

Statement map for Fundamental Theorem of ArithmeticAny finite list of primes whose product is n is a permutation of Mathlib's canonical prime-factor list for n. The pinned upstream declaration is Nat.primeFactorsList_unique. The exact checked statement is theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList.Mathematical readingAny finite list ofprimes whose product isn is a permutation ofMathlib's canonicalprime-factor list for n.Pinned declarationmathlib ·Nat.primeFactorsList_uniqueExact checked formtheoremNat.primeFactorsList_unique{n : ℕ} {l : List ℕ} (h₁: l.prod = n) (h₂ : ∀ p∈ l, Nat.Prime p) :l.Permn.primeFactorsListStatement map for Fundamental Theorem of ArithmeticAny finite list of primes whose product is n is a permutation of Mathlib's canonical prime-factor list for n. The pinned upstream declaration is Nat.primeFactorsList_unique. The exact checked statement is theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList.Mathematical readingAny finite list ofprimes whose product isn is a permutation ofMathlib's canonicalprime-factor list for n.Pinned declarationmathlib ·Nat.primeFactorsList_uniqueExact checked formtheoremNat.primeFactorsList_unique{n : ℕ} {l : List ℕ} (h₁: l.prod = n) (h₂ : ∀ p∈ l, Nat.Prime p) :l.Permn.primeFactorsList

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The theorem at a glance

Fundamental Theorem of Arithmetic — exact Mathlib form

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

Mathlib's selected declaration gives the uniqueness of prime-factor lists up to permutation. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual 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.

A prime list with product n has the same entries, including multiplicity, as n.primeFactorsList; only their order may differ. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual 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.

Complementary intuition

A mathematical landmark

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.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList
Selected evidence declaration text
theorem Nat.primeFactorsList_unique {n : ℕ} {l : List ℕ} (h₁ : l.prod = n) (h₂ : ∀ p ∈ l, Nat.Prime p) : l.Perm n.primeFactorsList

Claim boundary

No new theorem is claimed

This target indexes Mathlib's uniqueness theorem for prime-factor lists: any list of primes with product n is a permutation of n.primeFactorsList. It does not claim an ordered factorization, a factorization algorithm, a new proof, or a new mathematical result.

Provenance

Origin and evidence stay separate

Existing declaration
Nat.primeFactorsList_unique in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.fundamental-theorem-arithmetic.v001
Source handling
Verified upstream reference; no mirrored source package