Why are prime numbers considered special

Question

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

Answer 1

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.