Sylvester's Problem and Mock Heegner Points

TL;DR

This paper proves the existence of rational solutions to specific cubic equations involving primes congruent to 4 or 7 modulo 9, under the condition that 3 is not a cube modulo those primes.

Contribution

It establishes new results connecting prime congruences, cubic equations, and the properties of mock Heegner points, advancing understanding in number theory.

Findings

Solutions exist for x^3 + y^3 = p and p^2 under given conditions

Conditions on primes relate to the solvability of cubic equations

New links between prime congruences and rational points on cubic curves

Abstract

We prove that if $p \equiv 4,7 \pmod{9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb{Q}$.