Solve the Diophantine Equation $x^2 + 1 = 2y^4$ over $\mathbb{Z}$.
I have found few elementary solutions like $(1,1)$.
I have tried it with variable replacements. After solving it a bit it becomes clear that both $x$ and $y$ are odd. Evaluating it for $x = 2x_1+1$ and $y=2y_1+1$ I have reached:
$$2x_1^2 + 2x_1 + 1 = (2y_1 + 1)^4$$
or
$$x_1^2 + (x_1 + 1)^2 = (2y_1 + 1)^4$$
I don't know how to solve this further. Do you have any ideas, hints or techniques to solve this?
Thanks.