Definition:Odd Integer/Definition 3
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Definition
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
- Results about odd integers can be found here.
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Example $\text A.3$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.23$