**Title:** Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture**Abstract:** I will present a rank 0 and 1 p-converse theorem for CM elliptic curves defined over the rationals in the case where p is ramified in the CM field. This theorem has applications to two classical problems of arithmetic: it verifies Sylvester's conjecture from 1879 on primes expressible as a sum of two rational cubes and establishes Goldfeld's conjecture for the congruent number family, in particular settling the congruent number problem in 100% of cases. The proof relies on formulating and proving a new Iwasawa main conjecture, which in turn entails new methods arising from interplays between Iwasawa theory and relative p-adic Hodge theory on the infinite-level Shimura curve.