Definition:Odd Integer

(Redirected from Definition:Odd Number)

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.