Question 30
Prove that if $n$ and $p$ are integers, $1 \lt p \lt n$, then the number
$$
(2p+1)n^3 + 6n(1^2 + 2^2 + 3^2 + \dots + p^2)
$$
can be written as the sum of $2p + 1$ different perfect cubes.
$$
(2p+1)n^3 + 6n(1^2 + 2^2 + 3^2 + \dots + p^2)
$$
can be written as the sum of $2p + 1$ different perfect cubes.