64 is a square number () and a cube number (). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general?

Question

Homi · Accepted Answer

A number that is both a perfect square and a perfect cube must have its prime factors raised to powers that are multiples of six.

Step 1 — Understanding square and cube numbers

Let us consider a number, let's call it $N$ . We are looking for numbers $N$ that are both perfect squares and perfect cubes. This means $N$ can be written as the square of some whole number $a$ .

$N = a^2$

It also means $N$ can be written as the cube of some whole number $b$ .

$N = b^3$

For example, 64 is $8^2$ and also $4^3$ .

Diagram 1

Step 2 — Finding the general form

Let us think about the prime factors of $N$ . If $N$ is a perfect square, all the exponents in its prime factorization must be even numbers. For example, $36 = 2^2 \times 3^2$ . The exponents are 2 and 2, which are even. If $N$ is a perfect cube, all the exponents in its prime factorization must be multiples of 3. For example, $216 = 2^3 \times 3^3$ . The exponents are 3 and 3, which are multiples of 3.

For $N$ to be both a perfect square and a perfect cube, the exponents of its prime factors must be multiples of both 2 and 3. The smallest number that is a multiple of both 2 and 3 is their least common multiple (LCM).

$\text{LCM}(2, 3) = 6$

So, the exponents of the prime factors of $N$ must be multiples of 6. This means $N$ must be of the form $k^6$ for some whole number $k$ . Let us check if a number of the form $k^6$ is both a square and a cube.

We can write $k^6$ as a square:

$k^6 = (k^3)^2$

Here, $a = k^3$ . So, $k^6$ is a perfect square.

We can also write $k^6$ as a cube:

$k^6 = (k^2)^3$

Here, $b = k^2$ . So, $k^6$ is a perfect cube.

$\boxed{\text{Numbers of the form } k^6 \text{ are both squares and cubes.}}$

Step 3 — Listing specific numbers

Now we can find the first few numbers that fit this general form $k^6$ . Let $k$ be a whole number starting from 1.

For $k=1$ :

$N = 1^6$

$= 1$

We check: $1 = 1^2$ and $1 = 1^3$ .

For $k=2$ :

$N = 2^6$

$= 64$

We check: $64 = 8^2$ and $64 = 4^3$ .

For $k=3$ :

$N = 3^6$

$= 729$

We check: $729 = 27^2$ and $729 = 9^3$ .

For $k=4$ :

$N = 4^6$

$= (2^2)^6$

$= 2^{12}$

$= 4096$

We check: $4096 = 64^2$ and $4096 = 16^3$ .

For $k=5$ :

$N = 5^6$

$= 15625$

We check: $15625 = 125^2$ and $15625 = 25^3$ .

Answer

(i) The numbers that are both squares and cubes are: (ii) 1, 64, 729, 4096, 15625, and so on. (iii) The way to describe such numbers in general is that they are all numbers of the form $k^6$ , where $k$ is a whole number. This means they are perfect sixth powers.

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