I am trying to show that the equation: \begin{equation}x^4+py^4+p^2z^4=p^3w^4\end{equation} has no solutions.
Assuming there is a nonzero solution $(x_0,y_0,z_0,w_0)$, with $w_0$ minimal, then it must be that $p \, \vert \, {x_0}^4 \implies p \, \vert \, x_0$. Writing $x_0=px_1$, substituting this into the equation and simplifying: $$p^3{x_1}^4+{y_0}^4+p{z_0}^4=p^2{w_0}^4.$$ Hence, $p \, \vert \, y_0$. Continuing in this way, we find that $w_0=pw_1$, and so $(x_1,y_1,z_1,w_1)$ is a (nonzero) solution with $w_1<w_0$, a contradiction.
Does this constitute a proof by infinite descent?