Algebra, Geometry, & Number Theory Seminar

Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, asserts that every prime p = 4, 7, 8 mod 9 is the sum of two rational cubes. We discuss its recent resolution by Yin for primes p = 4, 7 mod 9, and by Burngale and Tian for the remaining case p=8 mod 9.The proofs are rooted in Heegner's seminal work and lie at the intersection of explicit complex multiplication and Shimura reciprocity, the Yuan–Zhang–Zhang generalization of the Gross–Zagier formula, auxiliary Rankin–Selberg constructions for Heegner points, and classical cubic descent along with its derived variant. They also rely on the resolution of the unbounded denominators conjecture, as well as recent rank-zero results towards the Birch–Swinnerton–Dyer conjecture for auxiliary CM elliptic curves.