# Category:Prime Decompositions

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This category contains results about **Prime Decompositions**.

Definitions specific to this category can be found in Definitions/Prime Decompositions.

Let $n > 1 \in \Z$.

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

\(\ds n\) | \(=\) | \(\ds \prod_{p_i \mathop \divides n} {p_i}^{k_i}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}\) |

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 $n$**.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Prime Decompositions"

The following 12 pages are in this category, out of 12 total.