Solutions to the equation $x^4+3y^4=z^2$ It was proven that the equation $x^4+2y^4=z^2$ has no non-trivial solutions in integers. What about the equation 
$x^4+3y^4=z^2$? It has a solution $x=1,y=1, z=1$. Can we find all solutions? 

Yes, the equation $x^4+3y^4 = z^2$ is birationally equivalent to an elliptic curve, hence we can find all its solutions.
However, if an easy proof there are infinitely many integer solutions with $\gcd(x,y)=1$ will suffice, then given an initial solution to,
$$x^4+dy^4=z^2$$
then, in general, further ones can be generated as,
$$S_n=\frac{x_n}{y_n}=\frac{-x^4+dy^4}{2xyz}$$
Example: Let $d=3$, and initial $x,y,z = 1,1,2$, then,
$$S_1 = \frac11\\
S_2 = \frac12\\
S_3 = \frac{47}{28}\\
S_4 = \frac{3035713}{6824776}
$$ 
such that,
$$47^4+3\times28^4=2593^2$$
and so on.
