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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.


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

  1. divisors,
  2. the divisors' arithmetic mean and
  3. 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

  1. What are the divisors of the numbers?
  2. 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

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