<s>[INST] Integers solutions to the equation $x^4 + 3y^4 = z^2$ I would like to find the integer solutions to the equation  $x^4 + 3y^4 = z^2$.
I know this has infinitely solutions, because I noticed if $x=y$, we have $4x^4 = z^2 \Rightarrow 2x^2 = z$ (for positive integers). So for instance $(1,1,2), (2,2, 8), ...$ are solutions.
How to find all the integer solutions to this equation?
 [/INST] The below equation has solution shown below:
$x^4 + 3y^4 = z^2$
$x=3m^2-n^2$
$y^2=4mn(3m^2+n^2)$
$z=(9m^4+18m^2n^2+n^4)$
RHS of $(y^2)$ is made a square at $(m.n)=(12,3)$ &
we get after removing common factors:
$(x,y,z)=(47,28,2593)$
Also for, $(m,n)=(1,1)$ we get:
$(x,y,z)=(1,2,7)$
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