Definition:Prime Number

Euclid's Definition


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

Equivalence of Definitions
These definitions are shown to be equivalent in Equivalence of Definitions of 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 $\N$.

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
 * One is not Prime


 * $1$ Divides all Integers
 * Integer Divides Itself