Henry Ernest Dudeney/Puzzles and Curious Problems/123 - Two Cubes/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $123$

Can you find two cube numbers in integers whose difference shall be a square number?

Thus the cube of $3$ is $27$, and the cube of $2$ is $8$,

but the difference, $19$, is not here a square number.

What is the smallest possible case?

The answer given by Dudeney is:

$8^3 - 7^3 = 512 - 343 = 169 = 13^2$

but it can be noticed that:

$2^3 - \paren {-2}^3 = 8 - \paren {-8} = 16 = 4^2$

which, because $8$ and $-8$ are both integers, is also a valid solution.