# Definition:Prime Decomposition

## Contents

## Definition

Let $n > 1 \in \Z$.

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

\(\displaystyle n\) | \(=\) | \(\displaystyle \prod_{p_i \mathop \divides n} {p_i}^{k_i}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}\) | $\quad$ | $\quad$ |

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 $n$.

## Examples

The prime decompositions for the first few integers are as follows:

$n$ Prime Decomposition of $n$ $1$ $1$ $2$ $2$ $3$ $3$ $4$ $2^2$ $5$ $5$ $6$ $2 \times 3$ $7$ $7$ $8$ $2^3$ $9$ $3^2$ $10$ $2 \times 5$ $11$ $11$ $12$ $2^2 \times 3$

## Also see

- Results about
**prime decompositions**can be found here.

## Linguistic Note

The UK English spelling of **prime factorization** is **prime factorisation**.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: 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