Integer solutions $d = \frac{ab}{a + b + 2\sqrt{ab}}$

Question

$$d = \frac{ab}{a + b + 2\sqrt{ab}} = \frac{ab}{(\sqrt{a} + \sqrt{b})^2}$$

What are the positive integer solutions?

The majority of solutions are when $a=b$, so that $a = 4d$.

Answer 1

$$b=m^2a$$ $$d=\frac{m^2a^2}{(1+m)^2a}=\frac{m^2a}{(1+m)^2}$$ $$a=(1+m)^2k,\quad b=m^2(1+m)^2k,\quad d=m^2k$$ For example,

$$m=1\rightarrow (a,b)=(4k,4k)$$ $$m=2\rightarrow (a,b)=(9k,36k)$$ $$m=3\rightarrow (a,b)=(16k,144k)$$ $$...$$