How many perfect squares can be written in the form $2{a^{2}}+3{b^{2}}$ with $a, b \in \mathbb{N}$?

Question

question

How many perfect squares can be written in the form $2{a^{2}}+3{b^{2}}$ with $a, b \in \mathbb{N}$?

my idea

I realised that the only solution is $(a,b)=(0,0)$

Let $a,b>0$ and we have to show that there are no perfect squares that can be written as $2{a^{2}}+3{b^{2}}$.

If $a=b \implies 5{a^{2}}$ should be a square number, but it's not, so we solved one case

If $a>b$ and $b>a$ are the cases left. i tried showing them by writing $a-b=k$ or $b-a=k$ but I get nowhere.