Fundamental Theorem of Arithmetic

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Theorem

For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquely up to the order in which they appear.


Proof

In Integer is Expressible as Product of Primes it is proved that every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes.

In Prime Decomposition of Integer is Unique, it is proved that this prime decomposition is unique up to the order of the factors.

$\blacksquare$


Also known as

This result is otherwise known as the unique factorization theorem.

However, this is the name of the equivalent result for the general Euclidean domain and it can be argued that the names are best kept separate.


Historical Note

The Fundamental Theorem of Arithmetic was first proved by Carl Friedrich Gauss in his Disquisitiones Arithmeticae in $1801$.


Sources