Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

Question

the question

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational.

the idea

A radical is rational only if the number below it is a square number. This means that for both of them to be simultaneously rational we get

$$a+b=x^2, a-b=y^2$$

From here I tried to take these two to a more general form that would lead us to infinitely many possibilities, but didn't get anything useful.

I hope one of you can help me! Thank you!

Answer 1

There are infinitely many odd squares $x^2$ and $y^2$ with $x>y>0$.

Let $a=\dfrac{x^2+y^2}2$ and $b=\dfrac{x^2-y^2}2$.