Mathematics Atlas

How Proof Is Made
Sign In
Text size
100%
Theme

Switch Atlas

The Inner World

The Civil World

The Natural World

Theorems

Fundamental Theorem of Arithmetic

Also Known As Unique Factorization Theorem
Number Theory

Citation Formats

General Reference

APA Style

BibTeX

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
Statement
Every integer greater than one can be represented uniquely as a product of prime numbers, up to the order in which the factors are written. 1
Proof Year
1801 1
Cross-Tradition Connections

In Branch

Number Theory, Branches of Mathematics
Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Arithmetic (Wikipedia)Opening section

Proved By

Carl Friedrich Gauss, Mathematicians
Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Arithmetic (Wikipedia)History section
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsHistory Topics: Prime numbers
Quote, 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.
View the Source
Fundamental Theorem of Arithmetic (Wikipedia)
Wikimedia FoundationHistory
Quote, History
Article 16 of Gauss's Disquisitiones Arithmeticae seems to be the first proof of the uniqueness part of the theorem.
View the Source
Fundamental Theorem of Arithmetic (Wikipedia)
Wikimedia FoundationProved By: Carl Friedrich Gauss, History section
Quote, 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.
View the Source
Fundamental Theorem of Arithmetic (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Opening section
Quote, In Branch: Number Theory, Opening section
every integer greater than 1 can be represented uniquely as a product of prime numbers
View the Source
Comments (0)
No comments yet. Be the first to share a thought.
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.

View At A Past Year

The atlas records no dated fact of its own for this entry, so there is no other year to choose.