Did Fermat actually have a proof?

Almost certainly not a correct general proof. The methods needed were developed only in the twentieth century, and any seventeenth-century argument would have relied on assumptions, such as unique factorization, that fail in the relevant rings.

How is an equation about powers related to elliptic curves?

A hypothetical solution can be packaged into the Frey elliptic curve; its arithmetic properties would contradict the modularity theorem, so the modularity of elliptic curves forces the original equation to be unsolvable.

Fermat's Last Theorem

Fermat's Last Theorem asserts that no three positive integers satisfy the equation a to the n plus b to the n equals c to the n for any exponent n greater than two — a claim that stood unproven for over three centuries until it was settled through the modularity of elliptic curves.

Definition

Fermat's Last Theorem is the statement that the equation x to the n plus y to the n equals z to the n has no solution in positive integers x, y, z whenever the integer exponent n is greater than two.

Scope

This topic covers the statement of Fermat's Last Theorem, its reduction to prime exponents and to the Fermat curve, Kummer's nineteenth-century progress using ideal numbers and regular primes, the Frey curve associated to a hypothetical solution, the epsilon conjecture proved by Ribet linking it to modularity, and Wiles's proof of the modularity of semistable elliptic curves that closes the argument.

Core questions

Why does it suffice to prove the theorem for prime exponents and for exponent four?
How far did classical methods, especially Kummer's theory of ideal numbers and regular primes, advance the problem?
How does the Frey curve turn a hypothetical Fermat solution into an elliptic curve with impossible properties?
How do Ribet's theorem and the modularity theorem combine to complete the proof?

Key theories

Kummer's regular primes: Kummer proved Fermat's Last Theorem for all regular prime exponents using ideal numbers, introducing the class group machinery of algebraic number theory in the process.
Frey curve and Ribet's theorem: A nontrivial Fermat solution would yield the Frey elliptic curve, which Ribet proved could not be modular; thus modularity of such curves would force Fermat's equation to have no solutions.
Modularity theorem (Wiles-Taylor): Wiles, with Taylor, proved that semistable rational elliptic curves are modular, contradicting the existence of the Frey curve and thereby proving Fermat's Last Theorem.

Clinical relevance

Though the theorem itself has no direct application, the proof's machinery — Galois representations, deformation theory, and modularity lifting — became core technology in the Langlands program and in the arithmetic-geometry methods that also inform elliptic-curve cryptography.

History

Fermat recorded the claim around 1637 in the margin of his copy of Diophantus, asserting a proof he never wrote down. Euler, Sophie Germain, and Kummer settled many cases over the next two centuries; Frey, Serre, and Ribet reduced it to modularity in the 1980s, and Wiles announced a proof in 1993, completed with Taylor in 1994 and published in 1995.

Key figures

Pierre de Fermat
Ernst Kummer
Ken Ribet
Andrew Wiles

Seminal works

wiles1995
wiles1995

Frequently asked questions

Did Fermat actually have a proof?: Almost certainly not a correct general proof. The methods needed were developed only in the twentieth century, and any seventeenth-century argument would have relied on assumptions, such as unique factorization, that fail in the relevant rings.
How is an equation about powers related to elliptic curves?: A hypothetical solution can be packaged into the Frey elliptic curve; its arithmetic properties would contradict the modularity theorem, so the modularity of elliptic curves forces the original equation to be unsolvable.