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?
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Please provide more context to the question, also show what you know (special cases) and what you don't. A good point to start is of course https://en.wikipedia.org/wiki/Pell%27s_equation ... it includes a discussion on some sligtly more general case https://en.wikipedia.org/wiki/Pell%27s_equation#Generalized_Pell's_equation – dan_fulea Jun 29 '21 at 15:02
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https://math.stackexchange.com/questions/526164/how-to-solve-an-equation-of-the-form-ax2-by2-cx-dy-e-0/829148#829148 – individ Jul 01 '21 at 05:57
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https://artofproblemsolving.com/community/c3046h1048219___2 https://artofproblemsolving.com/community/c3046h1048216__ – individ Jul 01 '21 at 08:12
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$a,b,c\in\mathbb{Z}$ as shown here – poetasis Aug 02 '21 at 01:54
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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
jjagmath
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