On the $8$ case of Sylvester Conjecture

TL;DR

This paper establishes a sufficient condition for primes congruent to 8 modulo 9, ensuring that either the prime or its square can be expressed as a sum of two rational cubes, advancing understanding of the Sylvester conjecture.

Contribution

It provides the first general result on the 8 case of the Sylvester conjecture, linking prime congruences to sums of rational cubes.

Findings

Identifies a sufficient condition for primes p ≡ 8 mod 9.

Shows either p or p^2 is a sum of two rational cubes under this condition.

Advances the theoretical understanding of the Sylvester conjecture.

Abstract

Let $p\equiv 8\mod 9$ 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.