Solutions to the equation $x^4+3y^4=z^2$

Question

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?

Answer 1

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.