Perfect powers expressible as sums of two cubes

Let n⩾3. This paper is concerned with the equation a3+b3=cn, which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer–Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density , and for a set of exponents n having natural density 1.