Paulo Ribenboim has written a book summarizing some of the results of the long attempt to prove Fermat's Last Theorem (we'll call it FLT from now on), focusing on results that were obtained entirely through elementary methods. While Ribenboim does not think such methods could lead to a proof of FLT, he finds much to admire in the new mathematics and ingenious arguments created along the way: "It would be an unforgivable error to let these gems sink into oblivion." Only in the last chapter of the book does he lift his self-imposed ban on non-elementary techniques, presenting an outline of key results in the final victory over FLT.
The phrase "for Amateurs" in the title should not at all be considered to have the same flavor as the phrase "for Dummies" that still seems to be a popular tag on many technical self-help books. It is also wise to keep in mind that "elementary" and "easy" are not synonymous. This book contains serious, densely presented mathematics that should be approached only by those with a love for the subject and a commitment to making their way through a thicket of formidable details. Nevertheless, it is a valuable record of an impressive body of mathematics created largely in pursuit of a single theorem.
In two pages before the first chapter, Ribenboim states The Problem, remarks it is sufficient to prove it for exponent 4 or an odd prime, and distinguishes between the traditional first case (solutions x, y, z not multiples of the odd prime p) and second case (p divides exactly one of x, y, and z). In the rest of this review, we shall write "Fermat equation" to refer to either xn + yn = zn or xn + yn + zn = 0, and "Fermat congruence" for either of those expressed as a congruence. The phrases "has solutions" and "is solvable" will be taken to include the condition that the solutions are integers and non-trivial, and FLT(n) will be shorthand for the truth of FLT for exponent n. p and q will be generic symbols for (usually distinct) odd primes.
In the first chapter, Special Cases, we find the usual parametric expressions giving all primitive Pythagorean triples, the remark that finding such triples amounts to determining which odd positive integers are sums of two squares, and necessary and sufficient conditions for a prime, and then for any natural number, to be a sum of two squares. Asking if the area of a Pythagorean triangle can be a square leads to the equation x4 - y4 = z2, which Fermat proved has no solutions. This result in turn shows that the Fermat equation with n = 4 has no solutions; Ribenboim gives a list of 20 authors of other proofs of that fact. A third section of this chapter demonstrates the unsolvability in Gaussian integers of x4 + y4 = z2, and a fourth shows that FLT holds for n = 3, with another list of authors of alternative proofs. A fifth section gives Gauss' proof that the case n = 3 of FLT has no solutions in the Eisenstein field of numbers of the form a + b*sqrt(-3), a and b rational, while section six presents a modern version of Dirichlet's proof of FLT(5). Section seven polishes off FLT(7), and the eighth section surveys the special cases n = 6, 10, and 14 (because the proofs are independent of the results for n = 3, 5, and 7). A brief appendix to the first chapter gives a failed infinite descent proof of FLT attributed to A. J. Lexell, and wonders if this could be the proof that Fermat thought he had.
The second chapter, 4 Interludes, prepares the reader for later developments by presenting material on p-adic valuations, cyclotomic polynomials, factors of binomials, and the resultant and discriminant of polynomials in two variables. Standing out from the highly technical background of most of the chapter are Legendre's 1808 expression for the exponent of the prime p in the prime factorization of a!, and an 1852 corollary of Kummer giving the exponent of p in a binomial coefficient.
Chapter III, Algebraic Restrictions on Hypothetical Solutions, gives some relations among the variables and exponent in Fermat's equation, constituting indirect proofs of FLT for some exponents.
Chapter IV, Germain's Theorem, discusses the establishment of the first case of FLT(p) for every prime p < 100. In Legendre's version of Sophie Germain's result, this involves deducing FLT(p) from two conditions: Fermat's congruence modulo some other odd prime q holds, and p is not congruent to a p-th power modulo q. For p < 100, primes q of the form mp + 1 (m = 2, 4, 8, 10, 14, or 16) can be found satisfying the conditions. In fact, Legendre had first-case results for all p < 197, after which the size of the numbers involved became a problem (in those pre-calculator days). The second section of the chapter presents a later 19th century result of E. Wendt on solutions of the Fermat congruence as above. The result uses the determinant W(n) of a cyclic n-by-n matrix of binomial coefficients, stating that the congruence is solvable if and only if q = 2kp + 1 divides W(2k). An appendix discusses the difficulty of determining when p and mp + 1 are both prime, and suggests a heuristic argument, based on Dirichlet's theorem and the prime number theorem, that there "should be" infinitely many primes with the desired properties.
The fifth chapter, Interludes 5 and 6, presents more background material for later use: Hensel's Lemma (1908) on p-adic roots of polynomials, then some results on second order recurrence sequences (such as the Fibonacci and Lucas sequences) including a generalization of Binet's formula.
The title of Chapter VI, Arithmetic Restrictions on Hypothetical Solutions and on the Exponent, pretty much delineates the content: congruences and divisibility relationships that must hold if the Fermat equation or congruence is satisfied. In some cases these give an indirect proof of FLT for certain exponents. Highlights include: Kummer's 1837 result that if the Fermat equation is satisfied for 2n (first case), then n = 1 (mod 8), with Dirichlet's Theorem providing the corollary that the set of odd primes p, for which the first case of FLT holds for 2p, is infinite; Terjanian's 1977 (!) elementary proof of the first case of FLT for even exponents; and Abel's conjecture of 1823, for which we still have no direct proof independent of Wiles' and Taylor's work, that no solution of Fermat's equation can be a prime-power.
Chapter VII, Interludes 7 and 8, contains identities involving powers of x + y + z, x + y - z, etc., and involving the Cauchy polynomials (x + y)n - xn - yn, especially in the case where y = 1. Many of the identities were used in early attempts to prove FLT or in early proofs of particular cases.
The eighth chapter, Reformulations, Consequences, and Criteria, presents first a collection of propositions that follow from, or are equivalent to, FLT. These range from Lebesgue's 1840 result that FLT(n) implies x2n + y2n = z2 has no solutions, to Frey's elliptic curve of 1986 that was instrumental in the eventual proof of FLT. Next we are given several reformulations of FLT, involving equations such as x(1 + x) = tn, (xy)m = x + y, and so forth, and including Quine's startling combinatorial formulation of 1989. Finally there is a miscellany of twenty or so conditions that are connected with, follow from, or imply FLT. These involve the Euler totient, the Moebius function, arithmetic progressions, the Legendre symbol, cubic congruences, and more.
Chapter IX, Interludes 9 and 10, is another collection of material needed in subsequent discussion, this time consisting of technical results from Galois theory involving Gaussian periods, Lagrange resolvents, and Jacobi cyclotomic functions.
In Chapter X, The Local and Modular Fermat Problem, we find first a proof that Fermat's equation with exponent p has solutions in q-adic integers (including when q = p), using Hensel's Lemma. Another key result considers the Fermat congruence modulo q, q not a divisor of n, with N(n, q) denoting the number of solutions of the congruence. In 1832 Libri proved that if N(p, q) = 0 for infinitely many q, then FLT is true for exponent p. We are also provided with upper and lower bounds for N(p, q), and a sufficient condition for solutions to exist, both due to L. E. Dickson in 1909. The third section of the chapter deals with the Hurwitz congruence a*xp + b*yp + c*zp = 0 (mod q), giving bounds and an existence condition that are counterparts to those for the Fermat congruence. The fourth and final section tackles the problem of the number of solutions of the Fermat congruence with exponent pm and modulus p(m + 1) and related congruences, arriving at relationships and expressions through a long string of technical lemmas.
In the final chapter, Epilogue, Ribenboim finally discusses FLT-related results that are not restricted to elementary methods. He begins with some approaches that "raised hopes for the proof" and "led to new research problems of independent interest." These are: Kummer's 1847 theorem on the truth of FLT for all regular primes; Wieferich's congruence 2(p - 1) = 1 (mod p2) which must hold if the first case of FLT is false, and which is satisfied by only two primes (1093 and 3511) smaller than 4*1012; a similar congruence showing that the first case of FLT is true for all p < 6.93*1017; a sieve theory proof that the first case holds for infinitely many exponents; Faltings' 1983 proof that there are at most a finite number of solutions to Fermat's equation for n > 3; and the abc conjecture, which implies FLT for sufficiently large exponents, and which Ribenboim says "should be very difficult [to prove]" and which "is presently the object of intense research." The second section of this chapter (titled Victory, or the Second Death of Fermat) celebrates the triumph of Wiles and Taylor and others, but also points out how much new mathematics was created in the long pursuit of that victory. In eight brief pages, Ribenboim recounts the now-familiar story of the Frey elliptic curves, the Shimura-Taniyama conjecture and Ribet's linking of it with FLT, and the work of Wiles with Taylor in tying everything together. The chapter concludes with 21 pages of text that are mainly bibliographic, listing 13 basic texts on elliptic curves and modular forms, 29 expository papers or books (only three of them published before 1993), and 23 research papers (mostly pre-1993) on the technical material underlying the Wiles proof, followed by six pages of material on incorrect or insufficient attempts at proof, and concluding with seven pages of general bibliography.
This book is a remarkable achievement. 350 years of work has been organized and condensed into a single volume so packed with detail that it sometimes seems as dense as a neutron star. Much of the mathematical content is presented by stating a definition or definitions (seldom with illustrative examples), followed by theorems with proofs. That rigorous style is always supplemented, however, with carefully selected words placing the material in a larger mathematical/historical context. Those who are mainly interested in the endgame of Wiles' proof should probably look elsewhere (as Ribenboim himself suggests in the Preface), perhaps to the excellent books by Simon Singh and Alf van der Poorten, or (as singled out by Ribenboim in the final pages of the book as "accessible for the courageous amateur") the 1994 Monthly article by Fernando Gouvêa. But those who realize that nearly every spectacular, seemingly individual mathematical feat is preceded and accompanied by the contributions of a host of unsung heroes will want to delve into this record of smaller victories along the way.
A rough count puts the number of citations at around 700. While many of these can be found in Ribenboim's 1979 13 Lectures on Fermat's Last Theorem (recently reprinted with an Epilogue on recent results, we are told), a great deal of ink has flowed in the twenty years since. Even allowing for considerable duplication in citing a work at the end of several sections, Ribenboim must have consulted or re-consulted 400 to 500 sources in putting together this book. It is a truly amazing feat of energy, persistence, and a non-amateur's love of the subject.
David Graves (firstname.lastname@example.org) is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in opera and history of astronomy as well as the usual run of mathematics courses.