Quartic Pell's equation - solving

Question

We are all aware of the fact that the equation $a^2-Nb^2=1$ ($N\in\mathbb{N_{\ge 0}}$, $N\neq k^2$ for some $k\in\mathbb{Z}$) has infinitely many solutions such that $a,b\in\mathbb{Z}$. But what about such equation as $a^{\color{red}4}-Nb^{\color{red}4}=1$. Can we also find infinitely many solutions to such equation (such that $a,b\in\mathbb{Z}$)?

Answer 1

Such an equation has only finitely many integer solutions. This is a special case of a classical (but still hard) 1909 theorem of Thue. See Wikipedia's article on Thue equations for further information.