On the $8$ case of Sylvester Conjecture

Abstract

Let $p\equiv 8\mathrm{mod}9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational cubes. This is the first general result on the $8$ case of the so-called Sylvester conjecture.

This is an improved version of the previous work. I get a much simpler but stronger result in this new version. Based on the suggestions of referee, I add or rewrite some of the proofs in order to make it easier to read. I also modify the computation mistakes and typos. This version is accepted to appear in Tran.AMS