[[Number theory MOC]]
# Fundamental theorem of arithmetic
The **fundamental theorem of arithmetic** is a consequence of [[Euclid's lemma]] which states 
that for any natural number there exists a unique prime factorisation when the factors are ordered by magnitude. #m/thm/num
That is, for any integer $n > 1$ there exists one and only one finite, increasing sequence of primes $(a_{i})_{i=1}^{\#a}$ such that $n = \prod_{i=1}^{\#a}a_{i}$.

> [!check]- Proof sketch
> The proof proceeds as follows:
> First, generalise [[Euclid's lemma]] to finite products.
> We can then show that given any two sequences of primes factorising some  integer $n > 1$,
> any element of one must be an element of the other. <span class="QED"/>

More generally, a ring in which this holds is called a [[Unique factorization domain]].

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