Definition:Prime Number
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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:
The list of primes starts:
- $2, 3, 5, 7, 11, 13, 17, \ldots$
This sequence is A000040 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
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 $\Bbb P$ 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 $\N$.
Euclid's Definition
As Euclid defined it:
(The Elements: Book VII: Definition $11$)
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 24$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 3.12$
- George E. Andrews: Number Theory (1971): $\S 2.2$: Definition $2.2$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 22$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 12$
- George F. Simmons: Calculus Gems (1992): Chapter $\text{B}.2$
- For a video presentation of the contents of this page, visit the Khan Academy.
