  We prove 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
$C_{k}$ to elliptic curves, together with the Mordell-Weil sieve, to rule out
the existence of rational points on $C_{k}$ for various $k$.
