The Diophantine Equation 3u2−2=v6 Original Research Article

The theorem of Delaunay–Nagell states that:If d is a cube-free integer>1,then the equation x3+dy3=1has at most one solution in non-zero integers x, y, and if such a solution exists then[formula]is either the fundamental unit of the field[formula]or its square, the latter occurring for only finitely many values of d. Investigation of these exceptionaldvalues has led to the equation of the title [5, 3.9], which has only finitely many solutions. We prove that the title equation has no integer solution other than u=v=1, which give the known valuesd=19, 20, 28, therefore there are no otherdvalues.