Problem 131

Prime cube partnership

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?

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These problems are part of
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