Solving constrained Pell equations

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Abstract
Consider the system of Diophantine equations x 2 − a y 2 = b x^2 - ay^2 = b , P ( x , y ) = z 2 P(x,y) = z^{2} , where P P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to the cases P ( x , y ) = c y 2 + d P(x, y) = cy^2 + d and P ( x , y ) = c x + d P(x, y) = cx + d , which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

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