Why are prime numbers considered special Why are prime numbers considered special because of the properties of having whole number factors of 1 and itself?
This seems rather arbitrary given the test is algorithmic and would compete with an infinite number of algorithmic tests resulting in a seemingly random selection pattern.
Hope i have expressed myself adequately 
 A: Mostly because any non-prime integer can be factored in a unique product of prime numbers, that is: They are the atoms of the natural numbers. The same idea of primality can be expanded with a few tweaks$^{[1]}$ for polynomials, Gaussian integers, Hurwitz integers, etc. 
In the case of polynomials, the factorization depends on the domain you're in $(\Bbb{R},\Bbb{Q},\dots)$ and the set of prime objects is different for each one of them. Thinking a little further, they are an interesting mathematical object which abstract the idea of mounting a certain object in which there is only one way to mount it, with certain rules, can be achieved.
An interesting and usually unusual theorem is the following:

Theorem (C. Cellitti,1914). Every $2\times 2$ matrix with integral elements can be written as a product of powers of 
$$\begin{pmatrix}
{1}&{1}\\ 
{0}&{1}
\end{pmatrix} \quad \quad \quad \begin{pmatrix}
{1}&{0}\\ 
{1}&{1}
\end{pmatrix}$$
And matrices of the form
$$\begin{pmatrix}
{a}&{0}\\ 
{0}&{1}
\end{pmatrix}$$
Where $a\in \Bbb{Z}$.

This will show you that there are other sources of factorization in a way almost akin to the natural numbers. Take a look at Weintraub's: Factorization: Unique and Otherwise.
$[1]:$ See here for prime elements and here for irreducible elements. In some domains, they are different things.  
