# Mock Heegner points and Sylvester's conjecture

## Meeting Details

Abstract: We consider the classical Diophantine problem of writing positive integers $n$ as the sum of two rational cubes, i.e.\ $n=x^3+y^3$ for $x,y \in {\mathbb Q}$. A conjecture attributed to Sylvester asserts that a rational prime $p>3$ can be so expressed if $p \equiv 4,7,8 \pmod{9}$. The theory of mock Heegner points gives a method for exhibiting such a pair $(x,y)$ in certain cases. We prove Sylvester's conjecture in the case that $p \equiv 4,7 \pmod{9}$ and $3$ is not a cube modulo $p$. This is joint work with Samit Dasgupta.