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Concept
Fundamental theorem of arithmetic
Summary
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,
The theorem says two things about this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.
The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique
(for example, ).
This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example,
The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers. This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between Fermat's statement and Wiles's proof.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements.
(In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic.
(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by infinite descent.
Proposition 32 is derived from proposition 31, and proves that the decomposition is possible.
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