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.
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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The Taniyama-Shimura-Weil conjecture states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This conjecture was proven by Wiles in 1994, using his proof of Fermat's Last Theorem.

The Modularity Theorem, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This theorem is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Taniyama-Shimura-Weil conjecture, which was proven by 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.

The Taniyama-Shimura-Weil conjecture, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This conjecture is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Modularity Theorem, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This theorem is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Taniyama-Shimura-Weil conjecture, which was proven by 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
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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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The Modularity Theorem states that every elliptic curve over the rational numbers is modular, meaning that it is associated with a modular form of weight 2. This theorem was first conjectured by Taniyama and Shimura in the 1950s and was later proven by Wiles in 1994.

The proof of the Modularity Theorem is quite technical and involves the use of complex analysis, algebraic geometry, and number theory. Wiles' proof relied heavily on the theory of modular forms, which are functions that are invariant under certain transformations.

The Modularity Theorem has far-reaching implications beyond the proof of Fermat's Last Theorem. It has been used to prove other important theorems in number theory, such as the Birch and Swinnerton-Dyer conjecture, which relates the order of the torsion subgroup of an elliptic curve to the rank of the Mordell-Weil group.

In summary, the Modularity Theorem is a deep and important result in number theory that has had a profound impact on the field. It was used by Wiles to prove Fermat's Last Theorem, and it has since been used to prove other important theorems in number theory.
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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.
user
The Taniyama-Shimura-Weil conjecture states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This conjecture was proven by Wiles in 1994, using his proof of Fermat's Last Theorem.

The Modularity Theorem, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This theorem is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Taniyama-Shimura-Weil conjecture, which was proven by 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.

The Taniyama-Shimura-Weil conjecture, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This conjecture is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Modularity Theorem, which was proven by Wiles in 1994, states that every elliptic curve over the rational numbers is modular, which means that it is associated with a modular form. This theorem is a key ingredient in Wiles' proof of Fermat's Last Theorem, which was also proven in 1994.

The Taniyama-Shimura-Weil conjecture, which was proven by 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
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