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Fundamental theorem of arithmetic
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. For example,
The theorem says two things about this example: first, that 1200 can 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, ).
Using the standard conventions for the product of a sequence (the value of the empty product is 1 and the product of a single factor is the factor itself), the theorem is often stated as: every positive integer can be represented uniquely as a product of prime numbers, up to the order of the factors.
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.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
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Fundamental theorem of arithmetic
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. For example,
The theorem says two things about this example: first, that 1200 can 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, ).
Using the standard conventions for the product of a sequence (the value of the empty product is 1 and the product of a single factor is the factor itself), the theorem is often stated as: every positive integer can be represented uniquely as a product of prime numbers, up to the order of the factors.
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.
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.