# Equivalence of Definitions of Odd-Times Odd Integer

## Theorem

The following definitions of the concept of **Odd-Times Odd Integer** are equivalent:

### 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 $n = x y$.

## Proof

### $(1)$ implies $(2)$

Let $n$ be an odd-times odd integer by definition 1.

Then by definition:

- $n > 1$
- $n$ is not prime.

Thus:

- $n = r s$

where $1 < r < p$ and $1 < s < p$.

As $n$ is odd:

- $2 \nmid n$

and so:

- $2 \nmid r$

and:

- $2 \nmid s$

So both $r$ and $s$ are odd integers greater than $1$ such that $n = r s$.

Thus $n$ is an odd-times odd integer by definition 2.

$\Box$

### $(2)$ implies $(1)$

Let $n$ be an odd-times odd integer by definition 2.

Then by definition:

- $n = r s$

where $r > 1, s > 1$ are odd integers.

Thus $n$ is not prime by definition.

As $r$ and $s$ are odd:

- $2 \nmid r$

and

- $2 \nmid s$

and so:

- $2 \nmid r s$

So $n = r s$ is odd.

As $r > 1$ and $s > 1$ it follows from Divisor Relation on Positive Integers is Partial Ordering that $n > 1$.

Thus $n$ is an odd-times odd integer by definition 1.

$\blacksquare$