  Using only elementary arguments, Cassels solved the Diophantine equation
$(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization
$(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was
considered by Zhongfeng Zhang who solved it for $k=2$, $3$, $4$ using
Frey-Hellegouarch curves and their Galois representations. In this paper, by
employing some sophisticated refinements of this approach, we show that the
only solution for $k=5$ is $x=z=0$, and that there are no solutions for $k=6$.
The chief innovation we employ is a computational one, which enables us to
avoid the full computation of data about cuspidal newforms of high level.
