There are some problems in high school Olympiad that ask to find integer solution 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 time, $(*)$ can transform into finding all square element in a Pell-equation's root sequence and look up the remainder for some modulo, and some time $(*)$ can also be solved by using infinite decent. However, none of those method is reliable because the associated sequence of $(*)$ is periodic for every mod so it can not be use if $(*)$ have solution, and sometime infinite decent 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 finite integer solutions (https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points). So are there any reliable way of craking these type of equation in elementary (high-school level) way  ?

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?

There are some problems in high school Olympiad that ask to find integer solution of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic form and quite often, $Q(0)= \pm 1$.Some time, $(*)$ can transform into finding all square element in a Pell-equation's root sequence and look up the remainder for some modulo, and some time $(*)$ can also be solved by using infinite decent. However, none of those method is reliable because the associated sequence of $(*)$ is periodic for every mod so it can not be use if $(*)$ have solution, and sometime infinite decent 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 finite integer solutions (https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points). So are there any reliable way of craking these type of equation in elementary (high-school level) way ?

There are some problems in high school Olympiad that ask to find integer solution 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 time, $(*)$ can transform into finding all square element in a Pell-equation's root sequence and look up the remainder for some modulo, and some time $(*)$ can also be solved by using infinite decent. However, none of those method is reliable because the associated sequence of $(*)$ is periodic for every mod so it can not be use if $(*)$ have solution, and sometime infinite decent 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 finite integer solutions (https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points). So are there any reliable way of craking these type of equation in elementary (high-school level) way ?