<s>[INST] Integer solutions for $x^4 + 4xy^3 = z^2$ Find all triplets $(x,y,z)$ of integers so that $$ x^4 + 4xy^3 = z^2. $$
What I've done:
Suppose $x=0$. Then we see $z=0$ and hence $(0,y,0)$ is a solution.
Suppose $y=0$. Then we see $x^4 = z^2$ and hence $(x,0,\pm x^2)$ is a solution.
Suppose $z=0$. Then we see $x^3=-4y^3$, which gives $x=y=0$ because otherwise the number of factors of $2$ on both sides cannot be equal. 
We may now assume $x>0$ and $z>0$.
I reasoned, if all three are even, $4$ must divide $z$, hence we may divide both sides by $16$, yielding a contradiction by infinite descent. Rewriting the equation as $4xy^3 = (z+x^2)(z-x^2)$ hasn't resulted in any good. Mathematica has found no solutions for $0<x<1000$ and $0<|y|<500$, so I think there are no more solutions. I haven't come really far yet, so any help is appreciated. Thanks.
 [/INST] I found an answer based on what MXYMXY posted earlier.
The crucial step is to observe that:
$$
(2x^2)(z-x^2)(z+x^2) = 2x^2(z^2-x^4) = 2x^2(4xy^3) = (2xy)^3.
$$
Note that from my analysis above we may conclude that all factors on the LHS are non-zero.
Also note that $$(2x^2)+(z-x^2) = z+x^2.$$ 
Let $d = \gcd(2x^2,z-x^2) = \gcd(z-x^2,z+x^2) = \gcd(z+x^2,2x^2)$ and observe that $d^3|(2xy)^3$, thus $d|2xy$. Now set $a = \frac{2x^2}{d}$, $b = \frac{z-x^2}{d}$, $c = \frac{z+x^2}{d}$ and $n =\frac{2xy}{d}$. Note that $a$, $b$ and $c$ are pairwise coprime. It follows that:
$$
abc = n^3 \quad \textrm{and} \quad a+b = c, 
$$
from which we conclude that $ab(a+b) = n^3$. Since $a$, $b$ and $a+b = c$ are pairwise coprime, we see that all three must be third powers, which is impossible by Fermat's Last Theorem.
This shows there are no further solutions.
</s>