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On some ternary quartic Diophantine equations. (English) Zbl 0216.04001

The author considers the equation (*) \(z^2 = U_1^2 + U_2 U_3\) where the \(U_i\) are polynomials in \(x, y\), vanishing at the origin and having integral coefficients. He shows that very slight further conditions on the \(U_i\) ensure that (*) has infinitely many solutions in integers \(x, y, z\). By putting \(z\pm U_i= pU_2/q, qU_3/p\), he reduces (*) to the binary quadratic equation (+) \(2pqU_1 = p^2 U_2 - q^2 U_3\). He chooses integers \(p, q\) so that (+) has infinitely many solutions, including \(x= y=0\).

MSC:

11D25 Cubic and quartic Diophantine equations
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