We show that the Diophantine equation of the title has, for n > 1, no solution in coprime nonzero integers x, y and z. Our proof relies upon Frey curves and related results on the modularity of Galois representations.

Under some hypotheses we show that the Diophantine equation (1) has infinitely many solutions described by a family depending on k + 2 parameters. Some applications of the main result are given and some special equations are studied.

We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the L1-norm of certain exponential sums.