# 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.

## 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.