## Prime Partner

#### Problem

Given that $p$ is an odd prime and $n$ is a positive integer, prove that there always exists a value of $n$ for which the expression $n^2 + np$ is a perfect square.

For example, when $p = 7$, $9^2 + 9 \times 7 = 144 = 12^2$.

Furthermore, prove that this value of $n$ is unique.

Problem ID: 291 (22 Sep 2006) Difficulty: 4 Star