Solve in $\mathbb N$ the following equation $3x^2+2x=y^2$
So first we look at the cases where one of them is $0$. I will prove that $ x= 0 \iff y=0$.First if $x=0 \implies y^2=0 \implies y=0$ and if $y=0 \implies x(3x+2)=0 \implies x=0 $ or $x=-\frac{2}{3}$ , but since $x \in \mathbb N \implies x=0$.Now we can continue with $x \ne 0 , y \ne 0$.The equation can be written $x(3x+2)=y^2 \implies y^2=k_1x$ and $y^2=k_2(3x+2)$ where $k_1,k_2 \in \mathbb N^*$ and also $k_1 \ne k_2$.From here I don't really know what approach to take. I managed to guess the solution $x=2 , y=4$,but I don't know how to prove it's the only one , or if there are more.If you have any idea or a solution, I'm here to listen. Thanks!