TY - JOUR

T1 - On certain diophantine equations of diagonal type

AU - Bremner, Andrew

AU - Ulas, Maciej

N1 - Funding Information:
The authors thank the referee for a careful reading of the paper, and for suggesting numerous improvements. The first author acknowledges with gratitude the hospitality of the Jagiellonian University, Kraków, for a short visit when the results presented in this paper were finalized; research of the second author was supported by Polish Government funds for science , grant IP 2011 057671 for the years 2012–2013.

PY - 2014/3

Y1 - 2014/3

N2 - In this note we consider diophantine equations of the form. a(xp-yq)=b(zr-ws),where 1p+1q+1r+1s=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 x4-y4=4(z4-w4).

AB - In this note we consider diophantine equations of the form. a(xp-yq)=b(zr-ws),where 1p+1q+1r+1s=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 x4-y4=4(z4-w4).

KW - Diagonal diophantine equation

KW - Quartic surface

KW - Zariski topology

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U2 - 10.1016/j.jnt.2013.09.008

DO - 10.1016/j.jnt.2013.09.008

M3 - Article

AN - SCOPUS:84887584311

SN - 0022-314X

VL - 136

SP - 46

EP - 64

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -