Fundamental theorem of arithmetic

Theorem (Fundamental theorem of arithmetic)

Every integer can be uniquely represented as a product of prime numbers. More formally, let be a non-zero integer. Then, there exist unique prime numbers and positive integers such that .

The uniqueness of the representation means that if where are prime numbers and are positive integers, then , and there exists a permutation such that and for all .

A different way to state this is that the prime factorization of an integer is unique up to the order of the (different!) prime factors:

Theorem (Fundamental theorem of arithmetic)

\forall n \in \mathbb{Z}{*} ~~ &\exists! \text{ prime }p{1},p_{2},\ldots,p_{s} \text{ and }l_{1}, \ldots, l_{s} \in \mathbb{N} \text{ such that} \

&p_{1}<p_{2}<\ldots<p_{s} ~\text{ and } ~n = p_{1}^{l_{1}}p_{2}^{l_{2}}\cdots p_{s}^{l_{s}} \end{align}

Examples

The following are prime factorizations: , , .