Integers that are sums of two rational sixth powers

TL;DR

This paper identifies the smallest positive integer that can be expressed as a sum of two rational sixth powers but not as a sum of two integer sixth powers, using elliptic curve morphisms and the Mordell-Weil sieve.

Contribution

It introduces a novel approach combining elliptic curve morphisms and the Mordell-Weil sieve to analyze sums of rational sixth powers.

Findings

164634913 is the smallest such integer.

Established non-existence of rational points on certain curves.

Applied elliptic curve techniques to number theory problems.

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 $k$.