# Geometric Numbers

There exists a mathematical concept called an
**arithmetic number**. You can check it out more in-depth from
this
wikipedia article about arithmetic numbers.

In short, a number $a \in \mathbb{Z}$ is called an arithmetic number if the mean of its divisors is also an integer.

## Example

For example, number $14$ is an arithmetic number. Let's prove this by checking if the conditions above apply.

Divisors of the number $14$ are numbers $1, 2, 3, 7$ and $14$. The mean of the divisors is

$$\frac{1 + 2 + 7 + 14}{4}=\frac{24}{4}=6.$$

The mean $6$ is an integer, so number $14$ is an arithmetic number.

## What are Geometric Numbers?

Spoilers! There is no such thing as a **geometric number** (at
least according to my quick Googling). So, let's invent them! For this we can
look back to arithmetic numbers. What are they made of?

For a number to be arithmetic we look at its

- divisors,
- the divisors' arithmetic mean and
- if the mean is an integer.

We can device an analogue for the geometric number. We can see that the
defining aspect of an arithmetic number is the
**arithmetic** mean. Maybe for a geometric number we could
calculate a geometric mean? Does such a thing exist? A geometric mean does
exist based on this
wikipedia article about the geometric mean. So, now we have the building blocks for a definition.

## Definition of a Geometric Number

A number $a \in \mathbb{Z}$ is a
**geometric number** if the geometric mean $g_m$ of
its divisors is an integer.

## Examples of Using the Definition

Let's take a look numbers $5$ and $9$. The questions that we need to answer are

- What are the divisors of the numbers?
- Are the geometric means of the divisors integers?

### Example 1

Let's answer the questions for the number $5$.

$5$ is a prime, so the only divisors are $1$ and $5$. Let's calculate the geometric mean of the divisors. There are two divisors, so the degree of the root is $2$.

The geometric mean is

$$g_m=\sqrt{1 \cdot 5}=\sqrt{5}=2.236\ldots.$$

Now we can see that the geometric mean is not an integer, so it follows that $5$ is not a geometric number.

### Example 2

The divisors of $9$ are $1,3,9$, so the degree of the root is $3$. Let's calculate the geometric mean

$$g_m=\sqrt[3]{1 \cdot 3 \cdot 9}=\sqrt[3]{27}=3.$$

Now the mean is an integer, so $9$ is an actual geometric number!

## Geometric Numbers in Teaching

Since geometric numbers don't acually exist they provide a good pedagogical tool for discovery in mathematics teaching. Here are some tasks to get started with geometric numbers

- Determine which of the numbers less than $25$ are geometric numbers.
- Which kind of numbers seem to be geometric numbers?
- Can you make any generalisations about which kind of numbers are geometric numbers?