# Category:Definitions/Prime Decompositions

This category contains definitions related to Prime Decompositions.

Related results can be found in Category:Prime Decompositions.

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 $n$**.

## Pages in category "Definitions/Prime Decompositions"

The following 3 pages are in this category, out of 3 total.