The guarantee that prime factorization is unique: every whole number greater than one breaks down into primes in exactly one way, aside from the order the factors are written in. Euclid's Elements contains results implying the pieces of this fact, but Carl Friedrich Gauss gave the first fully rigorous, general statement and proof in his 1801 Disquisitiones Arithmeticae, which is why the theorem is usually dated to him rather than to antiquity.
Facts
StatementEvery integer greater than one can be represented uniquely as a product of prime numbers, up to the order in which the factors are written. 2 Classification
Statement Form Connections
Associated With
Prime Number, Concepts The theorem states every integer factors into primes in an essentially unique way, so its statement is the prime number concept applied.
Additional Source MacTutor History of Mathematics ArchiveHistory Topics: Prime numbers
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Arithmetic (Wikipedia)Opening section
Proved By
Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Arithmetic (Wikipedia)History section
Sources
1. Fundamental Theorem of Arithmetic (Wikipedia)
Wikimedia FoundationHistory
Article 16 of Gauss's Disquisitiones Arithmeticae seems to be the first proof of the uniqueness part of the theorem.
Lead section, statement-form reference
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely as a product of prime numbers, up to the order of the factors.
Proved By: Carl Friedrich Gauss, History section
Article 16 of Gauss's Disquisitiones Arithmeticae seems to be the first proof of the uniqueness part of the theorem.
In Branch: Number Theory, Opening section
every integer greater than 1 can be represented uniquely as a product of prime numbers
View the Source 2. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsHistory Topics: Prime numbersQuote, History Topics: Prime numbers
Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
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