Sylvester’s conjecture

Sylvester’s conjecture predicts that every prime $p>3$ with $p\equiv4,7,8\pmod 9$ is a sum of two rational cubes. The record includes historical claims of a complete proof, but no independently assessed proof is supplied.

Known results

• A $3$-descent gives the expected rank pattern for the relevant elliptic curves.
• For $p\equiv4,7\pmod 9$ with $3$ not a cube modulo $p$, both $p$ and $p^2$ are sums of two rational cubes, with rank $1$ (2017).
• The unconditional $p\equiv4,7\pmod 9$ result is claimed in a 2026 preprint, but remains unverified.

October 2026 shorter proof of the $8$-case

Hongbo Yin gives a shorter proof of the already obtained $p\equiv8\pmod 9$ case, using ordinary traces and a Kronecker congruence for modular functions. This simplifies one case but does not settle the remaining conjectural picture.

Current status (as of October 2026): The $8$-case is reported as established and the $4,7$ cases are claimed in recent work, but the complete conjecture has no independently verified resolution in the retrieved record.