Prime Cube Partnership
There are some prime values, $p$, for which there exists a positive integer, $n$, such that the expression $n^3 + n^2p$ is a perfect cube.
For example, when $p = 19$, $8^3 + 8^2 \times 19 = 12^3$.
What is perhaps most surprising is that for each prime with this property the value of $n$ is unique, and there are only four such primes below one-hundred.
How many primes below one million have this remarkable property?