In the 19th century, Sylvester conjectured that every prime which is congruent to 4, 7, or 8 modulo 9 can be expressed as the sum of two rational cubes. In the 1950s, Selmer restated the conjecture as part of a more general study of rational points on elliptic curves. Later, the conjecture was seen to be a very special case of the conjecture of Birch and Swinnerton-Dyer. However, an unconditional proof remained elusive until Elkies announced a proof in the early 1990s for the case where $p$ is 4 or 7 modulo 9. We will describe how special values of modular forms allow one to attack Sylvester's conjecture and other similar problems concerning elliptic curves. We will conclude with current research, joint with John Voight, which uses these methods to study the remaining open case, where $p$ is 8 modulo 9.