A sextic diophantine chain and a related Mordell curve

Abstract

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain \( \phi(x_1,\,y_1,\,z_1)=\phi(x_2,\,y_2,\,z_2)=\phi(x_3,\,y_3,\,z_3)=k\) where \(\phi(x,\,y,\,z)\) is a sextic form defined by \(\phi(x,\,y,\,z)\) \(=x^6+y^6+z^6-2x^3y^3-2x^3z^3-2y^3z^3\) and \(k\) is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve \(y^2=x^3+k/4\). While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank \(\geq 6\) using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is \(\geq 6\).