Integers that are sums of two rational sixth powers

Abstract

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 $k$.

14 pages. Our main result is now unconditional