What is the fundamental theorem of arithmetic?

The fundamental theorem of arithmetic states that every integer greater than one is either a prime number or can be uniquely represented as a product of prime numbers, up to the order of the factors. This crucial property explains why the number one is not considered a prime, as including it would break uniqueness. The theorem also generalizes into other algebraic structures, though it fails in certain systems like algebraic integers.

What we know

• 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. [2, 3]
• The theorem is one of the main reasons why 1 is not considered a prime number, since including 1 would destroy unique factorization. [2]
• The theorem generalizes to other algebraic structures such as unique factorization domains, principal ideal domains, and Euclidean domains. [2]
• The theorem does not hold for algebraic integers, and this failure contributed to difficulties in proving Fermat's Last Theorem. [2]

Where to go next

• how it workedWhy is 1 not a prime number?
• how we knowHow do we know there are infinitely many primes?
• the bigger pictureWhat are unique factorization domains?
• what followedHow does public-key cryptography use prime numbers?
• compared withWhat is the fundamental theorem of algebra?
• an unexpected connectionHow did the failure of unique factorization impact Fermat's Last Theorem?The failure of unique factorization in rings of algebraic integers caused errors in many attempted proofs over hundreds of years before Wiles finally succeeded.

Sources

• [1]fundamental theorem of arithmetic (Wikidata Q670235) · CC0 1.0
• [2]Fundamental theorem of arithmetic · CC BY-SA 4.0
• [3]Prime number · CC BY-SA 4.0
• [4]Euclid's theorem · CC BY-SA 4.0
• [5]Fundamental · CC BY-SA 4.0