# 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

• Define special. Even numbers are also special in their own way. So are multiples of $7$, or perfect squares, or triangular numbers etc.
– dxiv
Oct 3, 2016 at 5:35
• If you think all algorithms are equally interesting, I don't know what to tell you... Oct 3, 2016 at 5:36
• I think one answer is that investigating the properties of prime numbers has led to a lot of beautiful math. It wasn't obvious originally that the study of prime numbers would turn out to be so fruitful. Oct 3, 2016 at 5:39
• I guess i was getting at the test for prime was rather arbitrary. Also the idea that one i NOT prime and 2 is the only even prime just sat uncomfortably with me but Oppa answered this well i think with prime factors. You cannot from my initial look make any whole number from factors of non-prime. at least the first ones so i guess the atoms of positive whole numbers sounds special indeed :-) Oct 3, 2016 at 22:52

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$^{}$ 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.

$:$ See here for prime elements and here for irreducible elements. In some domains, they are different things.

• Thanks, "They are the atoms of the natural numbers" I have seen this but it really works. Weird :-) Oct 3, 2016 at 22:53
• @Justin Did you see and understand the proof of the unique factorization? Oct 4, 2016 at 1:15
• Where do you find such interesting theorems? Oct 6, 2016 at 21:25
• @jm324354 Take a look at Eves': Elementary Matrix Theory. It is an interesting book because it has more theorems and ideas than proofs - but has a lot of references. I find this kind of book to be very useful if you want to make a quick map of the subject to explore the areas you find interesting later. Also, take a look of Eves' other books too, they are really worth reading. Oct 6, 2016 at 23:15