Definition:Prime Number

Definition 1
A prime number $p$ is a positive integer that has exactly two positive divisors.

Also see
Those two divisors of $p$ are $1$ and $p$, from:


 * $1$ divides all the integers
 * Integer Divides Itself.

The list of primes starts:
 * $2, 3, 5, 7, 11, 13, 17, \ldots$

Extension to Negative Numbers
The concept of primality can be applied to negative numbers as follows:

Comment
It follows from this that $1$ is not a prime number by this definition, as $1$ has only one positive integral factor, that is, $1$ itself.

The wording of this definition saves having to make a special case for $1$, which (for all sorts of reasons) is not considered to be a prime number.

Notation
Some authors use the symbol $\Bbb P$ to denote the set of all primes. This notation is not standard (but perhaps it ought to be).

The letter $p$ is often used to denote a general element of $\Bbb P$, in the same way that $n$ is often used to denote a general element of $\N$.

Also defined as
Some more advanced treatments of number theory define a prime as being either positive or negative, by specifying that a prime number is an integer with exactly $4$ integer divisors.

By this definition, a composite number is defined as an integer (positive or negative) which is not prime and not equal to $\pm 1$.

There are advantages to this approach, because then special provision does not need to be made for negative integers.

Also see

 * Prime Number has 4 Integral Divisors