Definition:Prime Number

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

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 $\N$.

uses $\varpi$ (a variant of $\pi$, despite its appearance) to denote a general set of primes.

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

 * Equivalence of Definitions of Prime Number
 * One is not Prime


 * Definition:Composite Number


 * Definition:Titanic Prime: a prime number with $1000$ digits or more
 * Definition:Gigantic Prime: a prime number with $10 \, 000$ digits or more

Generalizations

 * Definition:Prime Element of Ring