# Definition:Composite Number

## Definition

A **composite number** $c$ is a positive integer that has strictly more than two positive divisors.

That is, an integer greater than $1$ which is not prime is defined as composite.

In the words of Euclid:

*A***composite number**is that which is measured by some number.

(*The Elements*: Book $\text{VII}$: Definition $13$)

### Sequence of Composite Numbers

Definition:Composite Number/Sequence

## Extension to Negative Integers

The definition of a **composite number** can be extended to the negative integers, as follows:

A negative integer $n$ is composite if and only if $\left|{n}\right|$ is composite.

## Special Cases

$0$ is not considered **composite**.

$1$ is also a special case - it is neither prime nor **composite**.

All of the other positive integers are either prime or **composite**.

### Plane Number

A **plane number** is the product of two (natural) numbers.

In the words of Euclid:

*And, when two numbers having multiplied one another make some number, the number so produced is called***plane**, and its**sides**are the numbers which have multiplied one another.

(*The Elements*: Book $\text{VII}$: Definition $16$)

### Solid Number

A **solid number** is the product of three (natural) numbers.

In the words of Euclid:

*And, when three numbers having multiplied one another make some number, the number so produced is***solid**, and its**sides**are the numbers which have multiplied one another.

(*The Elements*: Book $\text{VII}$: Definition $17$)

## Also known as

Some sources refer to a **composite number** as a **factorable** or **factorizable** number.

For many reasons, one being that **composite** is of fewer syllables and therefore more economical to say, **composite** is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Examples

### Example: $8$

$8$ is a **composite number**, because:

- $8 = 4 \times 2$

and so has $1$, $2$, $4$ and $8$ as divisors.

Thus it has more than $2$ positive divisors, and hence is so classified.

## Also see

- Results about
**composite numbers**can be found**here**.

## Historical Note

The concept of classifying numbers as **prime** or **composite** appears to have originated with the Pythagoreans.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $3$: The Integers: $\S 12$. Primes - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 12$: Highest common factors and Euclid's algorithm - 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.2$ Prime numbers - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**composite number** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**factorable**:**1.** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.1$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**composite number** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**factorable**:**1.** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Primes - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**composite**