Question 51
We will call a positive integer $n$ cool if it is a perfect square and there exist positive integers $a$, $b$, $c$, $d$, different from each other, with $a \gt b$ and $c \gt d$, such that $n = a^3 - b^3 + c^3 - d^3$. For instance, $225$ is cool because $225 = 15^2$ and $225 = 7^3 - 5^3 + 2^3 - 1^3$. Show that:
a) $2025$ is cool;
b) there are infinitely many cool numbers.
a) $2025$ is cool;
b) there are infinitely many cool numbers.