# 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**.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\S 2.4$: Exercise $3$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 24$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 13$: The fundamental theorem of arithmetic