Problem 131

Problem 131

Prime cube partnership

There are some prime values, p, for which there exists a positive integer, n, such that the expression n3 + n2p is a perfect cube.

For example, when p = 19, 83 + 82×19 = 123.

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?


对于某些素数p,存在正整数n,使得表达式n3 + n2p是立方数。

例如,对于p = 19,83 + 82×19 = 123