There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $Q(0)= \pm 1$. Some times, $(*)$ can transform into finding all square elements in a Pell-equation's root sequence and look up the remainder for some modulus, and some times $(*)$ can also be solved by using infinite descent. However, none of those methods are reliable because the associated sequence of $(*)$ is periodic for every modulus so it can not be used if $(*)$ has solutions, and sometimes infinite descent is not possible. But $(*)$ is also equivalent to $dxQ(x) = y^2$ which is a typical elliptic curve equation and it is known to only have finitely many integer solutions (https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points). So are there any reliable ways of cracking these types of equation in elementary (high-school level) way?
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$\begingroup$ What is $d$? A constant? Additionally, a quadratic form is homogeneous, so it can't be $\pm 1$ at $0$, and I'm not sure how you turn it to $d x Q(x) = y^2$ as well. $\endgroup$– Daniel WeberCommented Sep 17, 2023 at 17:14
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$\begingroup$ sorry my mistake, $Q$ is a quadratic polynomial, if $z=x^2$ is a solution to $Q(x^2) = dy^2$ then $dz(Q(z))$ is also a perfect square, so just need to solve the later $\endgroup$– jackdeanCommented Sep 17, 2023 at 18:18
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$\begingroup$ @jackdean Would you be so kind as to provide some examples of the type of problems that you have in mind? $\endgroup$– José Hdz. Stgo.Commented Sep 18, 2023 at 14:57
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$\begingroup$ Certainly! finding all fibonacci number which also square for example, it's equivalent to solve $5x^4+4=y^2$ and substitute into the integral solution of the curve $x(5x^2+4) = y^2$ $\endgroup$– jackdeanCommented Sep 18, 2023 at 15:34
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