# Definition:Odd Integer

## Definition

### Definition 1

An integer $n \in \Z$ is **odd** if and only if it is not divisible by $2$.

That is, if and only if it is not even.

### Definition 2

An integer $n \in \Z$ is **odd** if and only if:

- $\exists m \in \Z: n = 2 m + 1$

### Definition 3

An integer $n \in \Z$ is **odd** if and only if:

- $x \equiv 1 \pmod 2$

where the notation denotes congruence modulo $2$.

### Euclid's Definition

In the words of Euclid:

*An***odd number**is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

(*The Elements*: Book $\text{VII}$: Definition $7$)

## Sequence of Odd Integers

The first few non-negative odd integers are:

- $1, 3, 5, 7, 9, 11, \ldots$

## Odd-Times Odd

Let $n \in \Z$, i.e. let $n$ be an integer.

### Definition 1

$n$ is **odd-times odd** if and only if it is an odd number greater than $1$ which is not prime.

### Definition 2

$n$ is **odd-times odd** if and only if there exist odd numbers $x, y > 1$ such that $n = x y$.

### Sequence

The sequence of **odd-times odd integers** begins:

- $9, 15, 21, 25, 27, \ldots$

## Examples

$-5$ is an odd integer:

- $-5 = 2 \times \paren {-3} + 1$

$-10$ is an odd integer:

- $17 = 2 \times 8 + 1$

## Also see

- Results about
**odd integers**can be found**here**.

## Historical Note

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

It was their belief that **odd numbers** (except $1$) are **male**, and **even numbers** are **female**.

A commentator on Plato used the term **scalene number** for an **odd number**, in correspondence with the concept of a scalene triangle. In a similar way an even number was described as **isosceles**.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**odd number**