Fermat's Last Theorem

For every integer n greater than 2, no positive integers x, y, and z satisfy xⁿ + yⁿ = zⁿ. Andrew Wiles proved it using methods connecting elliptic curves and modular forms.

Frey curve: An elliptic curve constructed from a hypothetical solution to a Fermat equation. It translates a supposed counterexample into the object at the heart of the proof.
Pierre de Fermat: A seventeenth-century French lawyer and mathematician known for contributions to number theory. He stated the theorem and claimed to possess a proof that has never been found.
Fermat's Last Theorem for n = 4: The case of Fermat's Last Theorem stating that no positive integers satisfy x⁴ + y⁴ = z⁴. Fermat's descent proves this case and reduces the general problem to prime exponents and four.
Taylor–Wiles method: A method for proving modularity by patching auxiliary data across carefully chosen prime levels. It repaired a gap in Wiles's initial argument and secured the theorem's proof.
Ribet's theorem: A theorem showing that a Frey curve from a Fermat counterexample would violate a form of modularity. It establishes the contradiction linking a Fermat solution to the modularity conjecture.
Fermat's marginal note: Fermat's handwritten claim, made in a book margin, that he had a proof too large to fit there. The note launched the centuries-long search for a proof.
Infinite descent: A proof method that derives an impossible infinite sequence of smaller positive integer solutions. Fermat used descent to prove the fourth-power case.
Modularity lifting theorem: A theorem that transfers modularity from one Galois representation to another under specified conditions. Wiles's proof developed lifting techniques that now support many modularity results.
Modularity theorem: The theorem that every elliptic curve over the rational numbers is modular. Its applicable case contradicts the nonmodularity implied by a hypothetical Fermat counterexample.
Leonhard Euler: An eighteenth-century mathematician who proved the case n = 3 of Fermat's Last Theorem. His proof was an early major advance beyond Fermat's own work.

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