Ashay A. Burungale

A proof of Sylvester's conjecture

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove th...