11D41: Higher degree equations; Fermat's equation
Unlike equations of degree at most four, there is comparatively little structure which may be imposed on the solutions of general Diophantine problems of degree five or greater. Some particular classes of problems (e.g. Thue equations) admit effective analysis, but for typical problems of this type it is difficult to determine the entire solution set.
In the 17th century, Pierre Fermat conjectured that the equation x^n+y^n=z^n had no solution in nonzero integers with n greater than 2. Known as Fermat's Last Theorem, the problem has little direct impact on broader fields of mathematics, but has precipitated considerable research of significant impact on number theory and algebraic geometry.
We confine our remarks to a short history of the Fermat equation.
The Fermat problem attracted considerable attention, and in addition to substantial progress by mathematicians of note, also engendered many attempts of dubious quality by amateurs. (It should be stressed that no elementary solution has been found, and that none is likely to exist.)
Fermat himself provided a proof for the case n=4; indeed he proved by induction (popularly: by "infinite descent") the slightly stronger result that no two fourth powers can ever sum to a perfect square. Fermat's more general conjecture was in fact no more than a personal note discovered after his death; it does not appear to be the case that he ever claimed publicly to have proven the conjecture true --- had he indeed been convinced that he had a proof, it would have been typical for him to have issued the conjecture as a challenge problem to other mathematicians and eventually to have shared a proof in his letters to others (publication in journals not yet having become the only modus operandi of mathematical research).
Euler proved the result for n=3, and Dirichlet and Legendre proved it for n=5.
Several significant advances were made during 1847. Lamé offered a proof (Comptes Rendues v.24 pp. 310-315; also Liouville's Jour. v.12 137-171) but Liouville and Kummer noted that this proof assumed unique factorization in certain number fields, which had not been proven (and in fact is false in some cases). Cauchy attempted to repair the proof of unique factorization in a series of papers in Comptes Rendues (vol. 25, pp 37, 46, 93, 132, 177, 242, 285, and especially 181); he was able to show that these proofs applied when a certain number-theoretic condition is satisfied (expressed in terms of Bernoulli numbers) in "case 1" (when n does not divide any of x, y, or z). Kummer was able to provide a proof valid also in "case 2" (when n divides xyz), in papers which appeared in the Berichte of the Akad. Wissen. Berlin (pp. 132-91 and 305-319. (The latter also appeared in Crelle's Jour. 40 (1850) 93-138 and Liouville's Jour. 16 (1851) 454-498). Again, these proofs only apply for certain n (those in which certain number fields have a class number not divisible by n). In subsequent years, Many refinements increased the set of exponents n for which the result was known to be true, but no proof valid for all n has ever been completed along these lines (indeed, no proof which could be shown to be true for infinitely many n was known until 1985, by Adleman, Heath-Brown, and Fouvry).
By the 1980s attention had shifted from algebraic number theory to algebraic geometry as the appropriate tool for the problem. Gerd Faltings' work on the Mordell conjecture implied in particular that for any n there were at most a finite number of solutions to the Fermat equation (up to scaling). Frey noticed in 1986 that a nontrivial solution to the Fermat equation would be related to an anomalous situation in elliptic curves; Serre clarified the conjectured connection, and Ribet proved this in
K. A. Ribet, "On modular representations of Gal(Qbar/Q) arising from modular forms", Inventiones Mathematicae, Vol. 100 (1990) pp 431-476.What Ribet proved is that given a nontrivial solution to the Fermat equation
a^n + b^n = c^n(in which we may assume n prime), the elliptic curve described by the equation
y^2 = x ( x - a^n ) (x + b^n )would be fairly well-behaved (technically: it would be "semistable") but would also be a counterexample to the Taniyama-Shimura conjecture (that all elliptic curves over the rational numbers are modular).
The Taniyama-Shimura conjecture is of great interest in elliptic curves and was not completely proved by 1994. However, Wiles has given a proof valid for most elliptic curves (the "semistable" ones); in particular, his proof is sufficient to prove that counterexamples of the Frey/Ribet sort cannot exist. This proves there are no nontrivial solutions to the Fermat equation. (Further work on the T-S conjecture led to improved results in 1999, first by Conrad, Diamond, and Taylor, and then a complete proof by those three and Breuil, announced in June 1999.)
Wiles first announced his proof in 1993 but technical difficulties showed that proof to be incomplete; a year later he managed a complete proof (actually using slightly simpler ideas) with the collaboration of Taylor for the particular portion which had invalidated his earlier attempt. The papers were published together, constituting one issue of the Annals of Mathematics. The citations are
Andrew Wiles, "Modular elliptic curves and Fermat's Last Theorem", Annals of Mathematics, Vol. 141 (1995) pp 443-551. Richard Taylor and Andrew Wiles, "Ring-theoretic properties of certain Hecke algebras", Annals of Mathematics, Vol. 141 (1995) pp 553-572.Extensions of this work have already appeared but do not substantially reduce the machinery needed to understand this work. Casual enthusiasts of number theory should be warned that this proof of Fermat's Last Theorem is unlikely to be comprehensible without the investment of several years' study of algebraic number theory and algebraic geometry.
Many suggestive but incomplete proofs of Fermat's conjecture have also been produced, and in many cases have led to valuable developments in mathematics; these are not discussed here but may be found in some of the documents indicated below.
There are some natural extensions of Fermat's conjecture which have been much studied.
More information on elliptic curves is available.
Parent field: 11D: Diophantine equations
As part of the historical record we have
In addition to the events surrounding Wiles' proof, there are a couple of other items of interest here.