Fermat's Last Theorem

Solved, 1994Arithmetic and number theory

Statement

For every integer $n > 2$ , the Diophantine equation $x^n + y^n = z^n$ has no solutions in positive integers $x, y, z$ .

Andrew Wiles announced a proof in June 1993 at the Isaac Newton Institute in Cambridge, establishing the Taniyama–Shimura–Weil modularity conjecture for semistable elliptic curves — which, by Ken Ribet's 1986 theorem on Gerhard Frey's curve, implies Fermat's Last Theorem. During peer review, Nick Katz identified a subtle gap in the Euler-system bound used to control a Selmer group. Working with his former student Richard Taylor, Wiles bypassed the obstacle by developing the Taylor–Wiles method for Hecke algebras and Galois deformation rings, completing the corrected proof in September 1994. The result was published in May 1995 as two companion papers in the Annals of Mathematics (one by Wiles, and a joint paper by Taylor and Wiles).

Posed by, 1637Pierre de Fermat Solved by, 1994Andrew Wiles

The Taylor–Wiles modularity-lifting machinery transformed algebraic number theory. In 2001, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended Wiles's work to prove the full Taniyama–Shimura–Weil modularity theorem for all elliptic curves over $\mathbb{Q}$ , and Chandrashekhar Khare and Jean-Pierre Wintenberger (2009) proved Serre's modularity conjecture, which directly implies Fermat's Last Theorem. Generalised Fermat equations $Ax^p + By^q = Cz^r$ are governed by the Darmon–Granville theorem in the hyperbolic regime $1/p + 1/q + 1/r < 1$ , while Beal's conjecture and the $abc$ conjecture propose far-reaching Diophantine generalisations.

References

Andrew Wiles (1995). Modular elliptic curves and Fermat's Last Theorem · DOI:10.2307/2118559
Richard Taylor, Andrew Wiles (1995). Ring-theoretic properties of certain Hecke algebras · DOI:10.2307/2118560
Kenneth A. Ribet (1990). On modular representations of Gal(\bar{Q}/Q) arising from modular forms · DOI:10.1007/BF01231195

History of the problem

1637Pierre de Fermat writes in the margin of his copy of Bachet's Latin edition of Diophantus's Arithmetica that a cube cannot be split into two cubes, nor any higher power $n > 2$ into two powers of the same degree, claiming to have a "truly marvellous proof" too large for the margin; only his proof for $n = 4$ by infinite descent survives.Pierre de Fermat
1770Leonhard Euler publishes a proof for the exponent $n = 3$ using arithmetic in $\mathbb{Z}[\sqrt{-3}]$ ; a gap in his implicit assumption of unique factorisation is later filled using Gaussian and Eisenstein integers.Leonhard Euler
1823Sophie Germain proves the first general criterion affecting infinitely many prime exponents: if $p$ and $2p + 1$ are both prime, then $x^p + y^p = z^p$ has no integer solutions in which $p \nmid xyz$ (Case I of Fermat's Last Theorem).Sophie Germain
1847Ernst Kummer introduces ideal numbers to restore unique factorisation of ideals in cyclotomic fields $\mathbb{Q}(\zeta_p)$ and proves Fermat's Last Theorem for every regular prime $p$ (primes not dividing the class number of $\mathbb{Q}(\zeta_p)$ ).
1984Gerhard Frey suggests at an Oberwolfach meeting that a hypothetical counterexample $a^p + b^p = c^p$ yields a semistable elliptic curve $y^2 = x(x - a^p)(x + b^p)$ whose discriminant properties are so extreme that it should fail to be modular; Jean-Pierre Serre formulates the precise $\varepsilon$ -conjecture needed to complete the link.
1986Ken Ribet proves Serre's level-lowering $\varepsilon$ -conjecture (published in 1990), establishing that Fermat's Last Theorem follows directly from the Taniyama–Shimura–Weil modularity conjecture for semistable elliptic curves.
1993After seven years of solitary work, Andrew Wiles announces a proof of the modularity conjecture for semistable elliptic curves in a series of three lectures at Cambridge in June; during refereeing, a gap is discovered in the Euler-system calculation.Andrew Wiles
1994Andrew Wiles and Richard Taylor repair the gap in September using the Taylor–Wiles patching method for Hecke algebras and release the completed manuscripts, which are published in the Annals of Mathematics in May 1995.Andrew Wiles

Prizes

Wolfskehl Prize, 100,000 Goldmark (DM 75,000 in 1997) — awarded