  We prove, assuming GRH, that $164634913$ is the smallest positive integer
that is a sum of two rational sixth powers but not a sum of two integer sixth
powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of
morphisms from $x^{6} + y^{6} = k$ to elliptic curves, together with the
Mordell-Weil sieve, to rule out the existence of rational points on $C_{k}$ for
various $k$.
