# Number Squared making Cube is itself Cube

## Theorem

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

Let $a^2$ be a cube number.

Then $a$ is a cube number.

In the words of Euclid:

If a number by multiplying itself make a cube number, it will itself also be cube.

## Proof

$a^2$ and $a^3$ are both cube numbers.

$a^3, m_1, m_2, a^2$

is a geometric sequence of integers for some $m_1, m_2 \in \Z$.

$a^2, m_3, m_4, a$

is a geometric sequence of integers for some $m_3, m_4 \in \Z$.

From If First of Four Numbers in Geometric Sequence is Cube then Fourth is Cube it follows that $a$ is a cube number.

$\blacksquare$

## Historical Note

This proof is Proposition $6$ of Book $\text{IX}$ of Euclid's The Elements.