How can you factorize n if you know that n + d^2 is a perfect square?

Question

I have managed to proove the first part but i got stuck on the second part of this excercise:

Let us demonstrate that for a secure RSA modulus $n = pq$, the difference $|p − q|$ should not be too small.

(a) Let $q − p = 2d > 0$. Prove that $n + d^2$ is a perfect square.
(b) Suppose that there exists a small integer $d$ such that $n + d^2$ is a perfect square. Show how $n$ can be factorized.

Answer 1

You know that $n+d^2$ is a perfect square, i.e. $n+d^2=m^2$ for some positive integer $m$. You can find $m$ by computing $\sqrt{n+d^2}$, which is easy. Then $n=m^2-d^2=(m+d)(m-d)$, so $q=m+d$ and $p=m-d$.