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Solving constrained Pell equations. (English) Zbl 0945.11027

Summary: Consider the system of diophantine equations \(x^2 - ay^2 = b\), \(P(x,y) = z^{2}\), where \(P\) 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 \(P(x, y) = cy^2 + d\) and \(P(x, y) = cx + d\), which arise when looking for integer points on an elliptic curve with a rational 2-torsion point.

MSC:

11Y50 Computer solution of Diophantine equations
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
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