Definition:Prime Decomposition

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Definition

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 $n$.


Multiplicity

For each $p_j \in \set {p_1, p_2, \ldots, p_r}$, 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 $n$.
the prime representation of $n$.

Some sources say that $n$ is expressed in standard form, but that term is used for something else on $\mathsf{Pr} \infty \mathsf{fWiki}$.


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

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