What properties do prime numbers exhibit?

[Faiq]

Mod note: moved from a homework section
What properties do prime numbers exhibit which can be used in proofs to define them?
Like rational numbers have a unique property that they can be expressed as a quotient of a/b.
Even numbers have a unique property of divisibility by 2 and thus they can be expressed as 2x.
Similarly are there any unique properties for prime numbers?

 

[fresh_42]

A prime number $p$ has the following property (definition): $p$ isn't a unit and if $p$ divides a product then it divides a factor of it.
$$p \, | \, ab ⇒ p \, | \, a ∨ p \, | \, b$$
In case of integers, the units are $±1$, so $p \neq ±1$.

 

[PeroK]

Homework Statement

What properties do prime numbers exhibit which can be used in proofs to define them?
Like rational numbers have a unique property that they can be expressed as a quotient of a/b.
Even numbers have a unique property of divisibility by 2 and thus they can be expressed as 2x.
Similarly are there any unique properties for prime numbers?

I would google "prime number" and browse until you're bored. You could start here:

http://mathworld.wolfram.com/PrimeNumber.html

 

[mfb]

There are way too many properties of prime numbers to list all of them.

 

[Faiq]

There are way too many properties of prime numbers to list all of them.

I am asking for properties that can help me represent a prime number when I am proving a statement

 

[micromass]

I am asking for properties that can help me represent a prime number when I am proving a statement

Then you're asking for a huge list. You need to narrow down your question.

 

[epenguin]

In #1 you quote unique (and in fact defining) properties of rational and even numbers as how they can be expressed. The defining property of prime numbers is how they can not be expressed. (Same as for irrational numbers.)