An extension to Pell's equation

Question

During my number theory seminary, I found this interesting problem and I didn't know how to solve it. Given Pell's equation $$x^2-3y^2=1,$$ where $x,y \in \Bbb N,$ show that there are infinitely many solutions such that $x-1$ is a perfect square.

Answer 1

If $a^2-3b^2=1$, then $$((a+3b)^2+1)^2-3((a+3b)(a+b))^2=1 .$$

We also know that one solution where $a=7$ and $b=4$ exists. Thus there exist infinite such solutions.