How do I prove that $x^4 + 4y^4 \neq z^2$ if $xyz \neq 0$ ? I was reading this paper https://arxiv.org/pdf/1311.1451.pdf (2.1) and am not able to follow the part of the proof on page 3.
I want to know this for a presentation I am doing on impossible diophantine equations.
Update: I edited some stuff that was commented about. Thank you.
On page 2, right after equations (2), he gets to $x_0^2=s_0^4-t_0^4.$ If you're willing to accept it as known that a (nontrivial) difference of fourth powers cannot be a square then there's the contradiction already. This fact was already known to Fermat and has an infinite descent proof; there are other approaches to showing it, and the paper you quote seems to go on and bring out such an approach. [I don't know why the author wasn't willing at that point to just quote the difference of fourth powers result.]