## Prime cube partnership

### Problem 131

Published on Friday, 10th November 2006, 06:00 pm; Solved by 4744; Difficulty rating: 40%There are some prime values, *p*, for which there exists a positive integer, *n*, such that the expression *n*^{3} + *n*^{2}*p* is a perfect cube.

For example, when *p* = 19, 8^{3} + 8^{2}×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?