The Taniyama-Shimura-Weil conjecture, which was proven by Andrew Wiles in 1994, establishes a deep connection between elliptic curves and modular forms. Using this connection, Wiles was able to prove Fermat's Last Theorem by showing that there are no solutions to the equation x^n + y^n = z^n for n > 2 in the context of elliptic curves. In other words, Wiles used the Modularity Theorem to prove Fermat's Last Theorem.
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