# Definition:Prime Decomposition

## Definition

Let $n > 1 \in \Z$.

From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form:

$n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$

where:

$p_1 < p_2 < \cdots < p_r$ are distinct primes
$k_1, k_2, \ldots, k_r$ are (strictly) positive integers.

This unique expression is known as the prime decomposition of $n$.

### Multiplicity

For each $p_j \in \left\{ {p_1, p_2, \ldots, p_r}\right\}$, its power $k_j$ is known as the multiplicity of $p_j$.

## Also known as

The prime decomposition of $n$ is also known as the prime factorization of $n$.

## Linguistic Note

The UK English spelling of prime factorization is prime factorisation.