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