Ribet's Theorem, proven by mathematician Ken Ribet in 1986, demonstrates that the Taniyama-Shimura-Weil conjecture implies Fermat's Last Theorem. This result was a crucial step in Andrew Wiles's eventual proof of Fermat's Last Theorem, as it linked the world of elliptic curves and modular forms to the centuries-old problem.

Ribet's Theorem

The Taniyama-Shimura-Weil Conjecture, now known as the Modularity Theorem, posits that every elliptic curve over the rational numbers is modular, meaning it can be associated with a modular form. This conjecture was a pivotal component in proving Fermat's Last Theorem and illustrates the deep connection between number theory and complex analysis.

Taniyama-Shimura-Weil conjecture

Fermat's Last Theorem states that there are no whole number solutions to the equation x^n + y^n = z^n for n greater than 2, a conjecture proposed by Pierre de Fermat in 1637 and famously proven by Andrew Wiles in 1994 using sophisticated techniques from algebraic geometry and number theory. This theorem remained unsolved for over 350 years and its proof marked a significant milestone in the field of mathematics, showcasing the deep connections between seemingly unrelated areas.

Fermat's Last Theorem

Andrew Wiles

Elliptic curves are smooth, projective algebraic curves with a group structure, used extensively in number theory and cryptography due to their rich mathematical properties and applications in secure communications. They provide a framework for defining operations such as addition and scalar multiplication, which underpin cryptographic protocols like Elliptic Curve Cryptography (ECC) that offer high security with smaller key sizes compared to traditional methods.

elliptic curves

Modular forms are complex analytic functions that are invariant under a specific group of transformations, and they play a crucial role in number theory, particularly in the theory of elliptic curves and the proof of Fermat's Last Theorem. They are characterized by their transformation properties and the presence of a Fourier expansion, which makes them a rich area of study in both pure and applied mathematics.

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