No integer solution of this Diophantine equation $$x^4+y^4+1=z^2$$ is known other than the trivial ones.
While I was reading a paper of Don Zagier, I realized that his idea on the Euler's sum of powers conjecture might be helpful to solving the equation. He observed that if $$z^4-x^4-y^4=R^2-QS$$where $Q,R,S$ are quadratic homogeneous equation over $\mathbb{Q}$, then $Q=0$ is tangent to $z^4=x^4+y^4$ at four points. Then the parameterization of $Q=0$ proves that $z^4(p,q)-y^4(p,q)-x^4(p,q)$ is a square, where $z,y,x$ are quadratic homogeneous equation of $p,q$.
Question: Can the same trick be applied to $x^4+y^4+z^4$? I tried to find a suitable $Q$ but I failed.