Prime Number - Fundamental Theorem of Arithmetic

The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every positive integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors. Primes can thus be considered the “basic building blocks” of the natural numbers. For example:

23244	= 2 · 2 · 3 · 13 · 149
	= 22 · 3 · 13 · 149. (22 denotes the square or second power of 2.)

As in this example, the same prime factor may occur multiple times. A decomposition:

n = p₁ · p₂ · ... · p_t

of a number n into (finitely many) prime factors p₁, p₂, ... to p_t is called prime factorization of n. The fundamental theorem of arithmetic can be rephrased so as to say that any factorization into primes will be identical except for the order of the factors. So, albeit there are many prime factorization algorithms to do this in practice for larger numbers, they all have to yield the same result.

If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.