Definition:Odd Integer

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Definition

Let $n \in \Z$ be an integer.

Definition 1

$n$ is odd if and only if it is not divisible by $2$.

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


Definition 2

$n$ is odd if and only if:

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


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:

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

This sequence is A005408 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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:

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

This sequence is A071904 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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