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. 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
, any element of one must be an element of the other.
More generally, a ring in which this holds is called a Unique factorization domain.