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 -