<s>[INST] Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$. 
Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

I think that $x^2 + 2xy + y^2$ and $x^2 + y^2$ are not consecutive squares since the difference is even. I think it has some relation with other squares like $(x+y)^2$ and $(x-y)^2$. 
How should I proceed? I would love some hints.
 [/INST] Suppose $x^2+y^2=a^2$ and $x^2-y^2=b^2$, then by multiplying them you get $x^4-y^4=(ab)^2$. This last equation has no non-trivial solution; see e.g. Solving $x^4-y^4=z^2$.
</s>