Definition:Prime Number

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

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


 * $1$ divides all the Definition:Integer|integers
 * Every Integer Divides Itself.

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

Equivalent Definition
$p$ is prime iff $\tau \left({p}\right) = 2$, where $\tau \left({p}\right)$ is the tau function of $p$.

Odd Prime
Every even integer is divisible by $2$ (because this is the definition of even). Therefore, apart from $2$ itself, all primes are odd.

So, referring to an odd prime is a convenient way of specifying that a number is prime, but not equal to $2$.

Composite
An integer greater than $1$ which is not prime is defined as composite.

Extension to Negative Numbers
The concept of primality can be applied to negative numbers by defining a negative prime to be of the form $-p$ where $p$ is a (positive) prime.

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.

See Prime Number has 4 Integral 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.

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.

Some authors use the symbol $\mathbb{P}$ 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 $\mathbb{P}$, in the same way that $n$ is often used to denote a general element of $\N$.