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:

\(\ds n\) \(=\) \(\ds \prod_{p_i \mathop \divides n} {p_i}^{k_i}\)
\(\ds \) \(=\) \(\ds {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}\)


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

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