For which $a,b,c$ is the diophantine quadratic equation $ax^2+bx+c=y^2$ soluble?

Question

Given $a,b,c\in\mathbb{Z}$, consider the quadratic equation $ax^2+bx+c=y^2$. Are there any general methods for deciding whether this equation has any integer solutions for $x,y$, given the coefficients?

Answer 1

Solve for $x$ to get $$x = \frac{-b\pm \sqrt{b^2-4ac+4ay^2}}{2a}$$ so you need $b^2-4ac+4ay^2$ to be a square, lets say $b^2-4ac+4ay^2 = z^2$.

If $a$ is negative it's easy to see that you need only a finite search. That's also the case if $a$ is a square.

The remaining cases are studied under the name of generalized Pell's equation