Definition:Odd Integer/Definition 3

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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:

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

Also see