On certain Diophantine systems with infinitely many parametric solutions and applications

Ulas, Maciej

doi:10.4064/aa145-3-7

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The Equation A(x1

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2014

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“…The equation a(y 4 1 − f 1 (X) 2 ) = b(y 4 2 − f 2 (X) 2 ) The same ideas as used in the preceding sections allow treatment of slightly more general equations. In a recent paper [14], inter alia, it is proved that for any pair of non-zero integers a, b and any positive odd n the diophantine equation…”

Section: The Equation A(xmentioning

confidence: 99%

On certain diophantine equations of diagonal type

Bremner

Ulas

2014

Journal of Number Theory

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Abstract. In this note we consider Diophantine equations of the form a(x p − y q ) = b(z r − w s ), where 1 p + 1 q + 1 r + 1 s = 1, with even positive integers p, q, r, s. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p, q, r, s) = (2, 6, 6, 6) we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for (p, q, r, s) ∈ { (2,4,8,8), (2,8,4,8)}. In the case (p, q, r, s) = (4, 4, 4, 4), we present some new parametric solutions of the equation x 4 − y 4 = 4(z 4 − w 4 ).

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Section: The Equation A(xmentioning

confidence: 99%