Find all the integral solutions of the equation
$x^4-10y^4=1$
I know how to solve when the power is 2. But I don't know how to solve this equation. One idea which I thought is to split it into two square factors and solve them simultaneously. But it is becoming complicated and I am not able to understand whether they will be all the solutions or not. Please provide a detailed solution if possible.
Hint (not a solution): Pell's equation $X^2-10Y^2=1$ has fundamental solution $(X,Y)=(19,6)$, thus all non-zero solutions are given by Pell's recurrence formula ${\displaystyle X_{k}+Y_{k}{\sqrt {10}}=(X_{1}+Y_{1}{\sqrt {10}})^{k},}$
$$ (19,6),(721,228),(27379,8658),(1039681,328776),(39480499,12484830),\ldots $$ Now show from Pell's formula, that no pair is of the form $(X,Y)=(x^2,y^2)$ for integers $x,y$.