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.

ABSTRACT In previous work, a periodic surface model for computeraided nano-design (CAND) was developed. This implicit surface model can construct Euclidean and hyperbolic nano geometries parametrically and represent morphologies of particle aggregates and polymers. In this paper, we study the characteristics of degree elevation and reduction based on a generalized periodic surface model. Method...