Find all $x,y,z\in\mathbb{N\_0}$ so that there exists a $k\in\mathbb{N}$ so that
$$4^x+4^y+4^z=k^2$$.

We can transform this problem to:

Find all $a,b\in\mathbb{N\_0}$ so that there exists a $t\in\mathbb{N}$ so that
$$4^a+4^b+1=t^2$$
where $a=x-z, b=y-z, k=2^z \cdot t$.

We see that $3|t^2 \Rightarrow 3|t$ and so we have that $a+b\equiv2 \mod3$.

Looking at $\mod 5$ we can see that both of $a$ and $b$ can't be both even at the same time. So one of them has to be odd.