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 $𝑛 > 1$ there exists one and only one finite, increasing sequence of primes $(𝑎 𝑖) # 𝑎 𝑖 = 1$ such that $𝑛 = \prod # 𝑎 𝑖 = 1 𝑎 𝑖$ .

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.

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

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Fundamental theorem of arithmetic

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