The Taniyama-Shimura-Weil conjecture is a significant hypothesis in number theory that connects elliptic curves and modular forms. It suggests that every rational elliptic curve is associated with a modular form, meaning that there is a deep relationship between these two areas of mathematics. This conjecture gained prominence when it was used to prove Fermat's Last Theorem by Andrew Wiles in the 1990s. Wiles showed that if the conjecture holds true, then Fermat's Last Theorem must also be true, leading to a breakthrough in understanding the connections between different mathematical concepts.