"Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos et generaliter nullam in in-infinitum ultra quadratum potestatem in duos eiusdem nominis fas est divider cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet." -Pierre de Fermat
The above statement translates as follows: It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain. In algebraic form, this means that the equation x n + y n = z n has no (non-trivial) integral solution (for x, y and z) for all powers of n greater than 2. (A non-trivial solution is one in which x, y and z are all non-zero.)
This statement is known as Fermat’s Last Theorem (FLT), which the French (lawyer-cum-) mathematician Pierre de Fermat wrote down sometime around 1637 in his copy of the book Arithmetica written by the ancient mathematician Diophantus of Alexandria (200-300 C.E.). Although this was not the last assertion that Fermat made in his life, it is known as Fermat’s Last Theorem because it was the last one that remained unproven for over three and a half centuries until Andrew J. Wiles of the University of Oxford (now Sir Andrew Wiles), who had been drawn to the problem when he was just 10 years old, proved it in 1994.
This year’s Abel Prize, widely regarded as mathematics’ Nobel Prize, has been awarded to Wiles for, to quote the citation by the Norwegian Academy of Science and Letters, “his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory”. For n = 2, the statement is nothing but the Pythagoras’ theorem: a 2 + b 2 = c 2 , where a and b are the lengths of the legs of a right-angled triangle and c is the hypotenuse. This, as we know, has many non-trivial integral solutions for a, b and c, the simplest and most interesting one being 3, 4 and 5. These solutions are called Pythagorean triples (a, b, c), and there are infinitely many of them. FLT says that there are no such solutions for n > 2. While the statement is accessible to every high-school student, it had proved to be one of the longest-standing unsolved problems in mathematics. FLT, as is clear from the citation, is a problem that belongs to the field of mathematics called number theory, which deals with arithmetic properties of numbers, and had attracted generations of mathematicians especially because of Fermat’s tantalising remark that he actually had a proof. FLT has the distinction of having the largest number of false proofs among all the conjectures in mathematics.
But, remarkably, given the large number of failures, many famous mathematicians were reluctant to even approach the problem suspecting its unprovability. For instance, after trying to prove FLT, Carl Friedrich Gauss (1777-1835), considered the greatest mathematician in the history of the subject, wrote: “Fermat’s Last Theorem, as an isolated proposition, has very little interest for me, for I could easily lay down a multitude of propositions, which one could neither prove, nor disprove.” The German mathematician David Hilbert (1862-1943) is supposed to have said this in 1920 to explain why he was not attempting to prove FLT: “Before beginning I have to put in three years of intensive study, and I don’t have that much time to squander on a probable failure.” On the basis of the famous Austrian mathematician Kurt Friedrich Godel’s incompleteness theorem in mathematical logic, which he himself proved in 1931, there was also a growing belief among mathematicians of the 20th century that perhaps the problem belonged to the category of problems that could neither be proved nor disproved. (Godel’s theorem states that in any comprehensive logical system, it is possible to make statements that can neither be proved nor disproved within that system. This theorem essentially is a formalisation in the language of mathematical logic of the ancient paradox known as the “liar’s paradox”, wherein a liar declares “Everything I say is false”, which leads to an obvious paradox.)
But all the failed attempts did result in the development of new areas such as algebraic number theory in the 19th century and proof of the “modularity theorem” in the 20th century following Wiles’ proof of FLT. Wiles’ proof showed two essential things. One, FLT did not belong to the realm of Godelian unprovable conjectures. Two, it was not an isolated proposition a la Gauss as the solution showed deep connections between apparently unrelated branches of number theory. The proof, which is over 150 pages long, also demonstrated that while it certainly could not have fit into the margin of the page where Fermat wrote his statement, he could not have had a proof at that time with the mathematical concepts and tools that would have been at his disposal in the 17th century. Wiles’ proof invokes concepts such as elliptic curves and modular forms in which developments were made only centuries later. As the Norwegian mathematician Arne Sletsjoe notes in his popular description of FLT, Wiles’ proof would have been highly inaccessible to even a brilliant mathematician like Fermat if he had the opportunity to read it.
The Norwegian Academy awards the Abel Prize every year in memory of the exceptional Norwegian mathematician Niels Henrik Abel (1802-1829) for contributions of extraordinary depth and influence to the mathematical sciences and for work that has contributed to raising the status of mathematics in society and stimulating the interest of children and young people in mathematics. “Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem,” notes the citation by the Abel Prize Committee. The prize carries a cash award of six million Norwegian kroner, which is about €600,000 or $700,000.
Wiles was born on April 11, 1953, in Cambridge, United Kingdom. He obtained his bachelor’s degree in mathematics at Merton College, Oxford, in 1974 and his doctoral degree at Clare College, Cambridge, in 1980. During 1977-80, he also was at Harvard University. After a brief period at the Institute for Advanced Study, Princeton, in 1981, Wiles became a professor at Princeton University. During 1985-86, he spent time at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette (near Paris), and at the Ecole Normal Superieure, Paris. From 1988 to 1990, he was a Royal Society Research Professor at Oxford before returning to Princeton. He rejoined Oxford in 2011 as a Royal Society Research Professor.
He has received a number of other major awards, notable among which are the Rolf Schock Prize (1995), the Ostrowski Prize (1995), the Wolf Prize (1995/6), the Royal Medal of the Royal Society (1996), the U.S. National Academy Award of Sciences in Mathematics (1996) and the Shaw Prize (2005). Wiles narrowly missed being awarded the Fields Medal, which is given only to mathematicians under 40. When he proved FLT, he was already 41. But, in lieu of it, the International Mathematical Union, for the first time ever in its history, presented him with a silver plaque for his achievement. He was awarded the inaugural Clay Research Award (1999). In 2000, he was conferred the knighthood.
As a 10-year-old in Cambridge, Wiles happened to read a book on Fermat’s Last Theorem at a local library. According to Wiles, he had been intrigued by the problem that he as a young boy could understand and yet it had remained unsolved for 300 years. “I knew from that moment that I would never let it go,” Wiles said. “I had to solve it.”
During his early career in mathematics, he did not actively attempt to prove FLT since it was generally believed that the theorem had proved to be too difficult and was probably unsolvable. But he did come to solve it 30 years later, in a way that can only be described as audacious as it had defied scores of outstanding mathematicians before him since Fermat wrote down his statement in the margins. Wiles chipped away at it for nearly seven years in total secrecy (except for confiding in his wife) before he cracked it.
Although Fermat had claimed that he had a general proof of his conjecture, he left behind details of his proof only for the special case of n = 4. It can be easily shown that once the case n = 4 is proven, the general proof for all n > 2 requires that the theorem be proved only for all odd primes (2 is the only even prime). In other words, it had only to be shown that Fermat’s equation x n + y n = z n had no integer solutions when n is an odd prime. But what characterises n = 3 and higher powers such that while the equation has many solutions for n = 2, it has none for the very next power? As Sletsjoe explains, if you consider numbers, say, up to 10,000, there are 2,691 sums of two squares, 100 squares and 42 numbers that are both a square and a sum of two squares. In contrast, when you consider cubes, there are only 202 sums of cubes, 21 cubes and none of these is both a cube and a sum of cubes. This property of being a cube and a sum of cubes becomes so rare already up to 10,000 that it is unlikely that any number would satisfy the requirement, and that is what FLT asserts.
Between 1637 and 1839, only the special odd prime cases of n = 3, 5 and 7 could be proved. While Leonard Euler is credited with having proved for n = 3 in 1770, there were others after him who provided independent proofs. Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet independently proved the case of n = 5 around 1825. Later others, including Gauss (1875, posthumously), also gave alternative proofs. Gabriel Lame settled the case of n = 7 in 1839. All proofs of specific powers followed essentially the same technique (called the method of infinite descent) with which Fermat had proved for n = 4. Interestingly, Dirichlet’s proof for n = 14 was published in 1832, before Lame’s proof for n = 7 because the method used was different.
However, as n increased, the methods and arguments used became increasingly complex, and it was becoming increasingly clear that the general proof could not be obtained by building upon the proofs various people had arrived at through the 19th and early 20th centuries for individual values of n or certain special classes of primes. In the latter half of the 20th century, computational approaches were pursued to prove the theorem for large values of n. Harry Vandiver used a Standards Western Automatic Computer in 1954 to prove FLT for all primes up to 2,521. Samuel Wagstaff extended this to all primes less than 125,000 by 1978. By 1993, FLT was computationally proved for all primes less than four million. However, proof of FLT for individual exponents up to howsoever large a number would never be equivalent to a general proof as there may be an even higher number X for which the theorem does not hold.
The real breakthrough advance that led to Wiles ultimately proving FLT came from unexpected areas of mathematics research. In the 1950s, two young Japanese mathematicians, Yutaka Taniyama and Goro Shimura, noted that the solutions to a type of equations called elliptic curves were remarkably similar to specific expressions of a class of functions called modular forms. (An elliptic curve is an equation of the form y 2 = x 3 + ax + b, where a and b are constants. Such equations appear in the study of elliptical orbits of planets. By the beginning of the 19th century, mathematicians, Abel, among others, began studying such equations in their own right, given their interesting properties. Modular forms, on the other hand, are more abstract mathematical constructs that exhibit a great deal of symmetry and nice analytic properties.)
Taniyama and Shimura concluded that the observed similarity could not just be a coincidence and that there must be a deep connection at a more fundamental level that produced two identical sequences of numbers in two apparently entirely disconnected mathematical areas. Then, they made this astounding conjecture in 1955: Every elliptic curve could be associated with its own modular form. This is called the Taniyama-Shimura-Weil (TSW) conjecture following the French mathematician Andre Weil’s detailed exposition of it about 10 years later. This was a surprising and bold conjecture that no one had any idea of how to go about proving.
In 1984, the German mathematician Gerhard Frey provided the first unexpected link between TSW and FLT. He asserted that if TSW was true, then FLT would follow as a consequence. Frey showed that if FLT was assumed to be false, one could set up a weird (semi-stable) elliptic curve that would have no modular form. This would be contrary to the TSW conjecture, which says that all elliptic curves are modular. Two years later, the American mathematician Ken Ribet proved Frey’s assertion. Frey’s and Ribet’s works implied conversely that if the TSW conjecture was true then FLT must also be true. The three-and-a-half-century-old theorem could now be rephrased in terms of the mathematics of elliptic curves and modular forms. Now, therefore, the only hurdle that needed to be crossed to prove FLT was to prove the TSW conjecture. But no one knew how to attack it.
Childhood dream Ribet’s proof in 1986 provided Wiles with the necessary incentive and trigger to return to try and prove FLT, which was his childhood dream. It was an amazing twist of fate that the two areas that Wiles had specialised in—elliptic curves, his PhD subject under John Coates, and modular forms—turned out to be precisely the areas that would now lead to the resolution of FLT.
Wiles made the rather unusual choice of working all alone on FLT rather than collaborating with someone. According to the Norwegian Academy’s background information on him, he was worried that since the problem was so famous the news he was working on it would attract too much attention and he would lose focus.
In 1993, after seven years of intense research work in utmost secrecy, Wiles believed that he had a proof. He decided to go public during a lecture series in Cambridge. However, he did not announce anything related to FLT beforehand, and the title of his lectures, “Modular Forms, Elliptic Curves and Galois Representations”, did not reveal anything either. But rumour had somehow spread around the mathematical community, and he gave his three lectures on consecutive days to a packed lecture theatre.
However, later that year, a referee checking his manuscript found an error. Wiles was devastated. He was still away from his childhood dream. But he did not lose heart. With redoubled vigour, he set out to fix the error for which he enlisted one of his former students, Richard Taylor. After a year’s work, Wiles found a way to correct the error. “I had this incredible revelation,” Wiles said in a BBC documentary, his eyes welling with tears. “It was the most important moment of my working life.” No gaps were found in the final manuscript of the proof that Wiles sent to the journal Annals in Mathematics , and it was published in 1995 under the title “Modular Elliptic Curves and Fermat’s Last Theorem”.
“The new ideas introduced by Wiles,” notes the citation, “were crucial to many subsequent developments, including the proof in 2001 of the general case of modularity conjecture by Christopher Breuil, Brian Conrad, Fred Diamond and Richard Taylor. As recently as 2015, Nuno Freitas, Bao V. Le Hung and Samir Siksek proved the analogous modularity statement over real quadratic number fields.”
Although it may sound technical, it may be pertinent here to quote from the famed Royal Society’s statement on him when he was elected to it, which reveals his amazing ability to bring in innovative ideas and methods to tackle apparently intractable problems in mathematics: “Andrew Wiles is almost unique amongst number-theorists in his ability to bring to bear new tools and new ideas on some of the most intractable problems of number theory. His finest achievement to date has been his proof, in joint work with Mazur, of the ‘main conjecture’ of Iwasawa theory for cyclotomic extensions of the rational field. This work settles many of the basic problems on cyclotomic fields which go back to [Ernst] Kummer, and is unquestionably one of the major advances in number theory in our times. Earlier he did deep work on the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication—one offshoot of this was his proof of an unexpected and beautiful generalisation of the classical explicit reciprocity laws of Artin-Hasse-Iwasawa. Most recently, he has made new progress on the construction of L-adic representations attached to Hilbert modular forms, and has applied these to prove the ‘main conjecture’ for cyclotomic extensions of totally real fields—again a remarkable result since none of the classical tools of cyclotomic fields applied to these problems.” Proving FLT is the peak of his achievements and the ultimate testimony to Wiles’ remarkable mathematical skill and ability.
Martin Bridson, head of the Mathematical Institute, University of Oxford, where Wiles currently works, said this about Wiles: “No individual exemplifies the relentless pursuit of mathematical understanding in the service of mankind better than Sir Andrew Wiles. His dedication to solving problems that have defied mankind for centuries, and the stunning beauty of his solutions to these problems, provide a beacon to inspire and sustain everyone who wrestles with the fundamental challenges of mathematics and the world around us. His work will inspire mathematicians and scientists for centuries to come. We are immensely proud to have Andrew as a colleague at the Mathematical Institute in Oxford.” In fact, the building in the Mathematical Institute where Wiles sits has been named in his honour.
Robert Bryant, the president of the American Mathematical Society, had this to say about Wiles: “Dr Wiles’ fundamental work in number theory has implications far beyond its deep consequences in pure mathematics, deepening our understanding of some of the most fundamental algorithms that underlie communications in our modern world and providing enormous benefits to our society and our world.”