ON A DIOPHANTINE EQUATION OF CASSELS

J.W.S. Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square. This amounts to finding all integer solutions to the Diophantine equation y^2=3x(x^2+2). We describe an alternative approach to solving not only this equation, but any equation of the type y^2=nx(x^2+2), with n a natural number. Moreover, we provide an explicit upper bound for the number of solutions of such Diophantine equations. The method we present uses the ingenious work of Wilhelm Ljunggren, and a recent improvement by the authors.