# What is a Fermat equation

In June of last year, around 1,000 scholars from all disciplines gathered in the auditorium of Göttingen University to watch Andrew J. Wiles (Fig. 2) from Princeton University (New Jersey) accept the Wolfskehl Prize (Spectrum of Science, August 1997 , Page 113). This award was given in 1907 to whoever would be the first to prove Pierre de Fermat's most famous theorem. It was endowed with a sum that originally had a purchasing power of around three million marks according to today's monetary value. As a result of government-enforced war loans in the First and the currency reform after the Second World War, only 7,500 marks remained in 1948. In the 49 years since then, the fund has grown again through interest and compound interest, so that in the summer of 1997 DM 75,000 was still available.

But the bottom line was something else: For Wiles, a childhood dream had come true and a decade of intense effort had come to an end. The scientists gathered in Göttingen also applauded him because his proof opened new, revolutionary paths for mathematics.

For his argument, which in the usual, taciturn expression, comprises at least 130 printed pages, Wiles had to take up and develop many modern ideas of his discipline. In particular, the visionary conjecture put forward in this century by the Japanese Goro Shimura and Yutaka Taniyama, which asserts a connection between algebraic geometry and complex analysis, had to be tackled. In proving them - in part - Wiles built a bridge between these large areas. From now on, knowledge in one will also be fruitful for the other, and mathematical fields of work that are further apart will also come closer to one another.

**The prince of amateurs**

The birth register of Beaumont-de-Lomagne, a small town in Gascony in south-west France, records the birth of a Pierre de Fermat on August 20, 1601. Most likely, however, it is not the author of the rebellious conjecture, because he died - as his son Clément-Samuel wrote - on January 12, 1665 in Castres at the age of 57. It can be assumed that Pierre, born in 1601, did not live long and that his parents gave his brother, who was born six years later, the same name. This was quite common with the high child mortality rate at the time. There is no certainty, because the town's church records for the years 1607 to 1611 have been lost. The historical Pierre de Fermat (Fig. 1) became a lawyer and made a career as a royal magistrate at the local parliament in Toulouse and as a justice of the peace at the Chambre de l'Édit de Nantes in Castres; this chamber was set up specifically to settle denominational disputes after the Edict of Nantes, which guaranteed religious freedom to the Huguenots in 1598. He devoted his little free time to his passion, mathematics. Although he only ran it on the side, he did essential preparatory work for probability theory as well as differential and integral calculus. Isaac Newton (1643 to 1727), one of the founders of modern analysis, admitted that his work was based on "Monsieur Fermat's method of drawing tangents". Above all, however, Fermat was a master of number theory, the study of whole numbers and their relationships with one another. He often wrote to other mathematicians about his results and challenged them to enter into intellectual competition. This and the peculiarity of almost never disclosing one's own arguments aroused considerable displeasure. René Descartes (1596 to 1650), inventor and namesake of (Cartesian) coordinate geometry, called Fermat a "Gascogner" (this - factually correct - designation is also a swear word in French meaning boastful or braggart). And the Englishman John Wallis (1616 to 1703), one of the pioneers of calculus, once referred to him as "that damn Frenchman". However, Fermat probably did not even disseminate his greatest challenge among specialist colleagues. He wrote it down when he was studying the "Arithmetic" of Diophantos of Alexandria around 1637 (around 250 AD), at the point where positive rational solutions of equation a

^{2}+ b

^{2}= c

^{2}to be discussed. According to the Pythagorean Theorem, this relationship applies to sides a, b and c of a right triangle. It has integer solutions like a = 3, b = 4, c = 5 or a = 5, b = 12, c = 13. There are an infinite number of such Pythagorean triples. But Fermat generalized the formula and considered the whole family of equations a

^{n}+ b

^{n}= c

^{n}, where n denotes any natural number greater than 2, and claimed that none of these equations have solutions in positive integers (or rational) numbers a, b, c. Now it seems very strange that there are no Fermat triples (i.e. solutions a, b, c of equation a

^{n}+ b

^{n}= c

^{n}) if there are an infinite number of Pythagorean triples. Not only that, Fermat believed he could prove his claim. In retrospect, it appears to be the malevolence of a genius that on the margin of this page of "Arithmetica" he wrote only the proverbial remark: "I have found truly wonderful evidence for this assertion, but this margin is too narrow to encompass. " Fermat wrote down such remarks several times, and after his death his son published an edition of the "Arithmetika" together with the marginal notes. As a result, all of these claims were proven, except for the one that subsequently became notorious as "Fermat's last theorem" (so it is by no means the last mathematical theorem of his life). Numerous researchers tried in vain, including Leonhard Euler (1707 to 1783), the greatest mathematician of the 18th century (Fig. 3). His failure did not leave him in peace, so that he asked a friend to investigate the Fermat family's house to see whether a piece of paper with decisive content might not have survived. For the case n = 4 there is a proof by Fermat himself as a marginal note elsewhere in the "Arithmetika". Euler found two proofs for the case n = 3. Sophie Germain (1776 to 1831; Fig. 4) - who corresponded under the pseudonym Monsieur Leblanc because of the prejudices against female mathematicians - achieved her first breakthrough in the 19th century (Spektrum der Wissenschaft, February 1992, page 80). She found a general criterion for the correctness of Fermat's assertion with regard to such exponents n, which are prime numbers with the property that 2n + 1 is also prime. An example is n = 5, because 2n + 1 = 11 is just as prime as 5 itself. Gustav Lejeune Dirichlet (1805 to 1859), later successor to Carl Friedrich Gauß in Göttingen, was able to use "Sophie's Theorem" as a student in 1825 Prove Fermat's last theorem for n = 5 (see also box on page 98).

**A childhood dream**

The doctor and mathematician Paul Wolfskehl (1856 to 1906) probably tried in vain for many years to find a comprehensive solution to the great riddle himself. Allegedly, this activity even discouraged him, who suffered from multiple sclerosis and was paralyzed, from thoughts of suicide. In his will, which came into force the year after his death, he donated 100,000 gold marks to whoever would be the first to be successful. This encouraged thousands of amateurs to attempt evidence. But hundreds of specialists, including first-class number theorists, tried again and again in vain. One of those who held back was David Hilbert (1862-1943), who is considered to be the greatest mathematician of the beginning of this century; When asked in 1920 why he never looked at the problem, he replied: "Before I could begin, I'd have to put three years of intensive study into it, and I don't have enough time to waste on a likely failure. " At least Fermat's last theorem was proven by 1993 for all exponents up to n = 4,000,000. But the goal of general proof still carried the number theorists in their hearts, albeit not as a concrete project, but more like a chemist's goal of the alchemists to make gold: as a foolish, romantic dream from bygone times. Children love romantic dreams. In 1963, in Cambridge, England, ten-year-old Andrew Wiles read the last sentence in a book from Fermat's public library and decided to prove it. His teachers advised against wasting time on the impossible, as did his lecturers later on. John Coates at Cambridge University finally succeeded in directing the interest of his PhD student Wiles to one of the main directions of modern mathematics: the fruitful theory of elliptic curves. These objects appear for the first time, in purely algebraic disguise, in the "Arithmetika" of Diophantos, which inspired Fermat. Little did Wiles suspect that these preliminary exercises would ultimately lead him back to Fermat's last sentence. Elliptic curves are not ellipses. They are so called because they are described by algebraic equations, the solutions of which are elliptic functions; and these in turn got their name because some special of them play a role in calculating the lengths of elliptical arcs. The equations describing elliptic curves are of the form y

^{2}= x

^{3}+ ax

^{2}+ bx + c, where a, b and c are integers and are chosen so that in the special case y = 0 there are three different solutions x. These are equations in the two variables x and y. They are called cubic or third degree because the highest power that occurs, in this case x

^{3}, is of grade 3. A common question in number theory is: How many rational points are there on an algebraic curve? That is, how many solutions x, y are there to the algebraic equation describing the curve, so that both coordinates x and y are whole or rational numbers (quotients of whole numbers)? This is easy to answer for equations of the first or second degree. First degree curves are straight lines and always have an infinite number of rational points. Second degree curves are conic sections; they either have zero or an infinite number of rational points, and it is not difficult to decide which is the case. On the other hand, on the hyperelliptic curves, more complicated objects, which are characteristically described by equations of degree 4 or higher, there are either no or only a finite number of rational points. The English mathematician Louis Joel Mordell (1888 to 1972) suspected this in 1922, and Gerd Faltings, who now works at the Max Planck Institute for Mathematics in Bonn, proved it in 1983 (Spektrum der Wissenschaft, September 1983, page 16) . But the elliptical curves are particularly difficult. You can have zero, finite, or infinite rational points, and it is far from easy to decide. Module arithmetic offers a helpful approach: You replace every number that appears in the cubic equation with its remainder when dividing by a prime number p. Furthermore, one understands all arithmetic operations in such a way that the result of an addition or multiplication should not be the usual one, but its remainder when divided by p. This modified version of the equation is called its analogue modulo p (see Spektrum der Wissenschaft, September 1996, page 80). This trick can be carried out with different prime numbers. This creates a whole series of simpler problems related to the original. For each prime number module p the number of solution pairs x, y is noted. The great advantage of module arithmetic is that the values of x and y must be integers and cannot be greater than p; this reduces the problem to something finite. In order to gain insight into the original infinite problem, mathematicians now observe how the number of solutions changes when p varies; and using this information, they create what is known as an L-series to the elliptic curve E:

In this infinite series the coefficient is a

_{p}the power p

^{-s}to be calculated from the number of solutions to the cubic equation modulo p if p is a prime number. If the index j of a

_{j}is a composite number, the coefficient a is calculated

_{j}from the corresponding values for the prime factors of j. Namely, the L series is created by multiplying an infinite product. The funny thing is that there are also L-series for completely different mathematical objects, the so-called modular shapes. Module forms initially have nothing to do with module arithmetic. It is a class of functions that depend on complex numbers. (Complex numbers are numbers of the form x + iy, where x and y denote real numbers and i, the imaginary unit, is the square root of -1.) The special thing about a modular form is that its independent variable x + iy is one can subject a large class of transformations, the so-called module transformations, without the module form changing its value (essentially). Something comparable can be found with the trigonometric functions such as sine and cosine: You can transform an angle j by adding any integer multiple 2πn of 2π to j without changing the value of the function: sin (j + 2πn) = sin j . Such a property is called symmetry; the trigonometric functions show them to a limited extent. Modular forms, on the other hand, surprise with an enormous degree of symmetry - so much so that the French polymath Henri Poincaré (1854 to 1912), when he discovered the first modular forms at the end of the 19th century, initially did not believe in them. He told his colleagues that for two weeks he had started looking for a mistake in his bills as soon as he woke up. Then he finally gave up and accepted that modular shapes are extremely symmetrical. Less than ten years before Wiles first encountered Fermat's theorem, the young Japanese mathematicians Goro Shimura and Yutaka Taniyama came up with an idea that later became a cornerstone in his proof. They believed that modular shapes and elliptic curves were profoundly related, even though they appeared to belong to distant areas of mathematics. After all, both have an L-series, although the rules for their derivation are different. The two Japanese now made the claim that for every elliptical curve there is a modular shape with the same L-series and vice versa. In Shimura's formulation in the early 1960s: "Every elliptical curve is modular." It was clear to the two mathematicians that the consequences would be extraordinary if they were right. In general, more is known about the L-series of a modular shape than that of an elliptic curve. So one could find out more about the L-series of an elliptical curve by detour via the associated modular form than by the direct, extremely laborious way. In addition, a bridge between two so far unrelated branches of mathematics generally uses both disciplines, as each benefits from the knowledge that the other has already acquired. Although no one found a way to prove Shimura and Taniyama's hypothesis, it had considerable impact. In the 1970s it became fashionable to assume it was correct and to derive new results from this assumption - although hardly anyone expected that it would be proven in this century. It is tragic that one of the authors did not live to see the spectacular success of this idea: On November 17, 1958, Yutaka Taniyama committed suicide.

**The missing link**

In August 1984 a conference on algebraic number theory took place in Oberwolfach in the Black Forest. Gerhard Frey (Fig. 6) from the University of Saarbrücken, now a professor in Essen, used this opportunity to share an idea with his colleagues that he had been brooding over for half a decade: a new approach to prove Fermat's Last Theorem. Often times, to test an assertion, mathematicians tentatively make the assumption that it is false and then examine the consequences of that assumption. In this case the claim is that the Fermat equation a

^{n}+ b

^{n}= c

^{n}have no solutions in natural numbers a, b and c. Suppose that is wrong. Then there would just be two nth powers a

^{n}and b

^{n}with n greater than 2, the sum of which is a third n-th power cn. But if there are numbers with this property, then - according to Frey's idea - they can also be used as coefficients of a special elliptic curve: y

^{2}= x (x-A) (x + B), where A = a

^{n}and B = b

^{n}is. One quantity that is routinely calculated for an elliptic curve is its so-called discriminant A.

^{2}B.

^{2}(A + B)

^{2}. Because A and B are solutions to Fermat's equation, this discriminant is an nth power: A

^{2}B.

^{2}(A + B)

^{2}= (a

^{2}b

^{2}c

^{2})

^{n}, because according to the assumption, A + B = a

^{n}+ b

^{n}= c

^{n}. So if Fermat's last theorem is wrong, there is an elliptic curve whose discriminant is an nth power.Conversely, a proof that the discriminant of an elliptic curve can never be an n-th power would also prove Fermat's last theorem. Frey saw no way of constructing such a proof. He considered it provable, however, that an elliptic curve whose discriminant is an n-th power cannot be modular - if it exists at all - that contradicts the conjecture of Shimura and Taniyama. Frey had thus sketched a chain of proof that was still missing two links: If it were to be possible, firstly, to prove the conjecture of Shimura and Taniyama and, secondly, that the elliptical curve y

^{2}= x (x-A) (x + B) is not modular, then it would be proven that an elliptic curve with these properties cannot exist. So there could be no solution to Fermat's equation and Fermat's last theorem would be proven. Many mathematicians tried the second link in the chain: to show that the elliptic curve y

^{2}= x (x-A) (x + B), now called the Frey curve, cannot be modular. Jean-Pierre Serre from the Collège de France in Paris and Barry Mazur from Harvard University in Cambridge (Massachusetts) made important contributions in this direction. Finally, in June 1986, one of us (Ribet; Fig. 7) constructed a complete proof of this claim. In the context of this article we want to limit ourselves to a few pointers. Ribet's proof makes substantial use of the fact that one can add two points on an elliptic curve (Spektrum der Wissenschaft, September 1996, p. 80). This addition is defined in the most abstract way and far from adding up as one learns it in elementary school: For any two points P1 and P2 on the curve one defines a third one, also on the curve, and calls it P.

_{1}+ P

_{2}. This definition is designed in such a way that the addition fulfills the usual calculation rules. In the case of the elliptical curves, it is geometrically defined (Fig. 7): Draw the - uniquely determined - straight line through P

_{1}and P

_{2}. It intersects the curve at a third point P.

_{3}. Its mirror image with respect to the x-axis is by definition the sum Q = P

_{1}+ P

_{2}. This addition rule can be applied to the infinitely many points of an elliptic curve; but it is particularly interesting because there are finite sets of points with the decisive property that the sum of every two points from the set also belongs to the set. This is a group in the mathematical sense of the word: a set of points to which certain simple rules apply. Finite groups can be constructed for an elliptic curve by going over to arithmetic modulo p with a prime p. It turns out that the property of being modular is inherited from an elliptic curve to these finite groups: If an elliptic curve is modular, then so is its analogues modulo p. Ribet has now shown that a special finite group to the Frey curve cannot be modular. Conversely, it follows from this that the curve itself is also not modular. Only one link was missing in the chain, namely the conjecture of Shimura and Taniyama. The proof of Fermat's Last Theorem, an exotic problem for centuries, was suddenly within reach.

**Seven years of secrecy**

Wiles, then a professor at Princeton University, New Jersey, did not fail. He worked on the problem for seven years without saying a word about it. Not only did he want to avoid the pressure of public attention, but also wanted someone else to snatch the glory from under his nose by using preliminary ideas. During this time, he alone told his wife about his obsession - on their honeymoon. For his large project, Wiles had to draw on many essential discoveries in modern number theory. Often they were not enough for his purposes, and he could not avoid creating new tools and techniques himself. He himself describes his experience as a journey through a dark, unknown large house: "You come into the first room and it is pitch black. You trip around and bump into the furniture, but gradually you learn where each piece is. Finally, after maybe six months, you will find the light switch. You turn it on and suddenly everything is lit. You can see exactly where you have been. Then you go into the next room and spend another six months in the dark. Such are these breakthroughs. Sometimes they come in a flash, sometimes they last a day or two. They are the culmination of a long wandering in the dark - and one cannot be had without the other. " However, Wiles did not need to prove the conjecture of Shimura and Taniyama in full generality. It suffices to show that a particular subclass of elliptic curves is modular, provided the Frey curve - if it existed - were included in that subclass. That didn't make things that much easier, however. The subclass chosen by Wiles is still infinitely large and includes the majority of the interesting cases. Wiles used the same techniques as Ribet - and many more. At this point, too, there must be indications. The challenge was to prove that every elliptic curve in the subclass is modular. To do this, Wiles used the group property of the points on the elliptic curves and a theorem by Robert P. Langlands of the Institute for Advanced Study at Princeton and Jerrold Tunnell of Rutgers University in New Brunswick, New Jersey. This theorem says that within each elliptic curve of the class Wiles studied there is a special group of points that is modular. This is a necessary, but not yet sufficient, condition for the assertion that Wiles actually wanted to prove. Because if the entire elliptic curve is to be modular, then all of these subsets must be modular. If you only know that for one of them - which in this case only has nine elements - that means very little. But Wiles used this small group as a first breach to unroll everything else from below, as it were: From groups with 9 elements he went to groups with 9

^{2}= 81 elements over, then to those with 9

^{3}= 729, and so on. If he could continue the process indefinitely, he would end up with an infinite number of finite groups. Their union would have an infinite number of elements; if he could then prove that it, too, is a group and modular, he would thereby have proven that the original group as a whole is modular. Wiles did this job through some kind of induction proof. He only had to show that if one of the groups is modular, then the same applies to the next larger one. It's like knocking over an infinite row of dominoes. One only has to see to it that one tipping one always brings the next one down; then all you have to do is toast the first one. Likewise, the evidence of modularity rattles off one group after the other. Finally, Wiles was convinced that the proof was complete and announced his result on June 23, 1993 at a meeting at the Isaac Newton Institute for the Mathematical Sciences, Cambridge. His secret research program seemed crowned with success, and the mathematician community was as surprised and enthusiastic as the international press. As the media circus rose, the usual official review process by the experts began. Pretty soon, Nicholas M. Katz of Princeton University discovered a fundamental and devastating flaw in one step of the argument. In his induction proof, Wiles had borrowed methodology from Victor A. Kolyvagin of Johns Hopkins University in Baltimore, Maryland and Matthias Flach of the California Institute of Technology in Pasadena. But now it seemed that their method was inapplicable in this particular example. The childhood dream had turned into a nightmare.

**The repair**

For the next twelve months, Wiles retired, discussing the error only with his former PhD student Richard Taylor. Together they tried to make his evidence viable after all and to go others that Wiles had previously rejected. On August 11, 1994 Wiles had to confess to an audience of 3,000 at the International Congress of Mathematicians in Zurich that he had not yet been able to close the gap in evidence (Spektrum der Wissenschaft, November 1994, page 126). He and Taylor were about to admit defeat and publish the erroneous result so that others could try to improve it.

Then, on September 19, 1994, the decisive enlightenment came. Several years earlier, Wiles had considered an entirely different approach based on a theory by the Japanese mathematician Kenkichi Iwasawa, who taught at Princeton, but had made no headway with it and had given it up. But now he realized that the very reason why the method of Kolyvagin and Flach had failed was responsible for the success of the approach via the Iwasawa theory.

About his reaction to this discovery, he reports: "It was so indescribably beautiful - so simple and so elegant. The first evening I went home and slept on it. The next morning I checked everything again, and then I went downstairs and said yes my wife: 'I've got it. I think I've found it.' It was so unexpected that she thought I was talking about a children's toy or something and she said, 'Got what?' I said, 'I fixed my evidence. I did it.' "

For Wiles, the Wolfskehl Prize was the end of an obsession that lasted more than 30 years: "After solving this problem, it is certainly a feeling of freedom," he confessed. "I was so obsessed with this problem that I thought about nothing else for eight years - from getting up to going to bed. This very special odyssey is now over and my soul has come to rest."

Wiles will no doubt not retire; the Wolfskehl Prize is hardly likely to be enough for that. He will continue to do math and leaves no doubt that this will fill him up; only - the magic of the childhood dream is gone.

There is still enough to be done. In particular, it is generally believed that Wiles' proof is far too complicated and modern for Fermat to have meant when he wrote his marginal note. Either Fermat was mistaken and what he had in mind was flawed, or - hard to believe, but conceivable - a simple and astute proof is still waiting to be discovered.

**Bibliography**

Yutaka Taniyama and His Time: Very Personal Recollections from Shimura. By Goro Shimura in: Bulletin of the London Mathematical Society, Volume 21, Pages 186 to 196, 1989.

From the Taniyama-Shimura Conjecture to Fermat's Last Theorem. By Kenneth A. Ribet in: Annales de la Faculté des Sciences de l'Université de Toulouse, Volume 11, Issue 1, pages 115 to 130, 1990.

Modular Elliptic Curves and Fer-mat's Last Theorem. By Andrew Wiles in: Annals of Mathematics, Volume 141, Issue 3, Pages 443-551, May 1995.

Ring Theoretic Properties of Certain Hecke Algebras. By Richard Taylor and Andrew Wiles in: Annals of Mathematics, Volume 141, Issue 3, Pages 553-572, May 1995.

Notes on Fermat's Last Theorem. By A. J. van der Poorten. Wiley Interscience, 1996.

Fermat's last sentence. By Simon Singh. Published in spring 1998 by Hanser, Munich.

Comments on Diophantus by Pierre de Fermat. By Max Miller. Ostwald's Classics of Exact Sciences No. 234. Teubner, Leipzig 1932.

13 Lectures on Fermat's Last Theorem. By Paulo Ribenboim. Springer, New York 1979.

About Wiles' proof of the Fermat conjecture. By Gerhard Frey in: Mathematische Semesterberichte, Volume 40, Pages 177 to 191, 1993.

Fermat's Last Theorem and Modern Arithmetic. By Kenneth A. Ribet and Brian Hayes in: American Scientist, Volume 82, Pages 144-156, 1994.

Number Theory as Gadfly. By Barry Mazur in: American Mathematical Monthly, Volume 98, pages 593-610, 1991.

Modular Forms and Fermat's Last Theorem. Edited by Gary Cornell, Joseph H. Silverman, and Glenn Stevens. Springer, New York 1997.

From: Spektrum der Wissenschaft 1/1998, page 96

© Spektrum der Wissenschaft Verlagsgesellschaft mbH

#### This article is contained in Spectrum of Science 1/1998

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