712. Let $a$ be any natural number. Find some representation of the number $a^{3}$ as the difference of two perfect squares.
$\triangle$ Let $x$ and $y$ be such integers that

$$
x^{2}-y^{2}=a^{3}, \quad(x+y)(x-y)=a^{3}
$$

For the last equality to hold, it is sufficient that the following two equalities hold:

$$
\left\{\begin{array}{l}
x+y=a^{2} \\
x-y=a .
\end{array}\right.
$$

This means that if the numbers $x$ and $y$ satisfy this system of equations, then they also satisfy the equation $x^{2}-y^{2}=a^{3}$ (but not vice versa). To solve the system, we add and subtract its equations:

$$
2 x=a^{2}+a, \quad 2 y=a^{2}-a, \quad x=\frac{a(a+1)}{2}, \quad y=\frac{a(a-1)}{2}
$$

Thus, the equality holds:

$$
a^{3}=\left[\frac{a(a+1)}{2}\right]^{2}-\left[\frac{a(a-1)}{2}\right]^{2} \cdot \mathbf{A}
$$