Let k be an even integer such that k is at least 2. We give
a (natural) density result to show that for almost all d at least 2,the equation (x+1)^k + (x+2)^k + ... + (x+d)^k = y^n with n at least 2, has no integer solutions (x,y,n). The proof relies upon some Galois theory and group theory, whereby we deduce some interesting properties of the Bernoulli polynomials.
This is joint work with Samir Siksek (University of Warwick).