Fermat's Enigma

by Simon Singh

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean. G. H. Hardy

Some are influenced by the love of wealth while others are blindly led on by the mad fever for power and domination, but the finest type of man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher for although no man is completely wise in all respects, he can love wisdom as the key to nature’s secrets.

According to Pythagoras, numerical perfection depended on a number’s divisors (numbers that will divide perfectly into the original one). For instance, the divisors of 12 are 1, 2, 3, 4, and 6. When the sum of a number’s divisors is greater than the number itself, it is called an “excessive” number. Therefore 12 is an excessive number because its divisors add up to 16. On the other hand, when the sum of a number’s divisors is less than the number itself, it is called “defective.” So 10 is a defective number because its divisors (1, 2, and 5) add up to only 8. The most significant and rarest numbers are those whose divisors add up exactly to the number itself, and these are the perfect numbers. The number 6 has the divisors 1, 2, and 3, and consequently it is a perfect number because 1 + 2 + 3 = 6. The next perfect number is 28, because 1 + 2 + 4 + 7 + 14 = 28.

perfect numbers are always the sum of a series of consecutive counting numbers. So we have

Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1. That is to say,

Pythagoras’s theorem provides us with an equation that is true of all right-angled triangles and that therefore also defines the right angle itself. In turn, the right angle defines the perpendicular and the perpendicular defines the dimensions—length, width, and height—of the space in which we live. Ultimately mathematics, via the right-angled triangle, defines the very structure of our three-dimensional world.

At this point it is important to note that, although this theorem will forever be associated with Pythagoras, it was actually used by the Chinese and the Babylonians one thousand years before. However, these cultures did not know that the theorem was true for every right-angled triangle. It was certainly true for the triangles they tested, but they had no way of showing that it was true for all the right-angled triangles that they had not tested. The reason for Pythagoras’s claim to the theorem is that it was he who first demonstrated its universal truth.

Bertrand Russell would comment on this apparent oxymoron: “How dare we speak of the laws of chance? Is not chance the antithesis of all law?”

The Library was the idea of Demetrius Phalaerus, an unpopular orator who had been forced to flee Athens, and who eventually found sanctuary in Alexandria. He persuaded Ptolemy to gather together all the great books, assuring him that the great minds would follow. Once the tomes of Egypt and Greece had been installed, agents scoured Europe and Asia Minor in search of further volumes of knowledge. Even tourists to Alexandria could not escape the voracious appetite of the Library. Upon entering the city, their books were confiscated and taken to the scribes. The books were copied so that while the original was donated to the Library, a duplicate could graciously be given to the original owner. This meticulous replication service for ancient travelers gives today’s historians some hope that a copy of a great lost text will one day turn up in an attic somewhere in the world. In 1906 J. L. Heiberg discovered in Constantinople just such a manuscript, The Method, which contained some of Archimedes’ original writings.

Euclid believed in the search for mathematical truth for its own sake and did not look for applications in his work. One story tells of a student who questioned him about the use of the mathematics he was learning. Upon completing the lesson, Euclid turned to his slave and said, “Give the boy a penny since he desires to profit from all that he learns.” The student was then expelled.

π can never be written down exactly, because the decimal places go on forever without any pattern. A beautiful feature of this random pattern is that it can be computed using an equation that is supremely regular: By calculating the first few terms, you can obtain a very rough value for π, but by calculating more and more terms an increasingly accurate value is achieved. Although knowing π to 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom, this has not prevented computer scientists from calculating π to as many decimal places as possible. The current record is held by Yasumasa Kanada of the University of Tokyo, who calculated π to six billion decimal places in 1996.

In order to prove that could not be written as a fraction Euclid used reductio ad absurdum and began by assuming that it could be written as a fraction. He then demonstrated that this hypothetical fraction could be simplified. (Simplification of a fraction means, for example, that the fraction can be simplified to by dividing top and bottom by 2. In turn can be simplified to , which cannot be simplified any further and therefore the fraction is then said to be in its simplest form.) Furthermore, Euclid showed that his simplified fraction, which still was supposed to represent , could be simplified not just once but over and over again an infinite number of times without ever reducing to its simplest form. This is absurd because all fractions must eventually have a simplest form, and therefore the hypothetical fraction cannot exist. Therefore cannot be written as a fraction and is irrational.

For Pythagoras, the beauty of mathematics was the idea that rational numbers (whole numbers and fractions) could explain all natural phenomena. This guiding philosophy blinded Pythagoras to the existence of irrational numbers and may even have led to the execution of one of his pupils. One story claims that a young student by the name of Hippasus was idly toying with the number , attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e., that is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’s insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’s argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

Appropriately for a problem-solver, the one detail of Diophantus’s life that has survived is in the form of a riddle said to have been carved on his tomb: God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, He clothed his cheeks with down; He lit him the light of wedlock after a seventh part, and five years after his marriage He granted him a son. Alas! late-born wretched child; after attaining the measure of half his father’s full life, chill Fate took him. After consoling his grief by this science of numbers for four years he ended his life.

Fermat was not interested in writing a textbook for future generations: He merely wanted to satisfy himself that he had solved a problem. While studying Diophantus’s problems and solutions, he would be inspired to think of and tackle other related and more subtle questions. Fermat would scribble down whatever was necessary to convince himself that he could see the solution and then he would not bother to write down the remainder of the proof. More often than not he would consign his inspirational jottings to the bin, and then move on to the next problem.

One of Fermat’s discoveries concerned the so-called friendly numbers, or amicable numbers, closely related to the perfect numbers that had fascinated Pythagoras two thousand years earlier. Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand, the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220. The pair 220 and 284 was said to be symbolic of friendship. Martin Gardner’s book Mathematical Magic Show tells of talismans sold in the Middle Ages that were inscribed with these numbers on the grounds that wearing the charms would promote love. An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac. Early theologians noted that in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, was an expression of Jacob’s love for Esau.

Fermat noticed that 26 is sandwiched between 25 and 27, one of which is a square number (25 = 52 = 5 × 5) and the other is a cube number (27 = 33 = 3 × 3 × 3). He searched for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. After days of strenuous effort he managed to construct an elaborate argument that proved without any doubt that 26 is indeed the only number between a square and a cube. His step-by-step logical proof established that no other numbers could fulfill this criterion. Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it?

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet. I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain. This was Fermat at his most infuriating. His own words suggest that he was particularly pleased with this “truly marvelous” proof, but he had no intention of bothering to write out the detail of the argument, never mind publishing it.

All prime numbers (except 2) can be put into two categories: those that equal 4n + 1 and those that equal 4n – 1, where n equals some number. So 13 is in the former group (4 × 3 + 1), whereas 19 is in the latter group (4 × 5 – 1). Fermat’s prime theorem claimed that the first type of primes were always the sum of two squares (13 = 22 + 32), whereas the second type could never be written in this way.

“The Devil and Simon Flagg” the Devil asks Simon Flagg to set him a question. If the Devil succeeds in answering it within twenty-four hours then he takes Simon’s soul, but if he fails then he must give Simon $100,000. Simon poses the question: “Is Fermat’s Last Theorem correct?” The Devil disappears and whizzes around the world to absorb every piece of mathematics that has ever been created in order to prove the Last Theorem. The following day he returns and admits defeat: “You win, Simon,” he said, almost in a whisper, eyeing him with ungrudging respect “Not even I can learn enough mathematics in such a short time for so difficult a problem. The more I got into it the worse it became. Non-unique factoring, ideals—Bah! Do you know,” the Devil confided, “not even the best mathematicians on other planets—all far ahead of yours—have solved it? Why, there’s a chap on Saturn—he looks something like a mushroom on stilts—who solves partial differential equations mentally; and even he’s given up.”

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigor should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W. S. Anglin

The problem began by asking, What is the square root of one, ? The obvious answer is 1, because 1 × 1 = 1. The less obvious answer is –1. A negative number multiplied by another negative number generates a positive number. This means –1 × –1 = +1. So, the square root of +1 is both +1 and –1. This abundance of answers is fine, but then the question arises, What is the square root of negative one, ? The problem seems to be intractable. The solution cannot be +1 or –1, because the square of both these numbers is +1. However, there are no other obvious candidates. At the same time completeness demands that we must be able to answer the question. The solution for Bombelli was to create a new number, i, called an imaginary number, which was simply defined as the solution to the question, What is the square root of negative one? This might seem like a cowardly solution to the problem, but it was no different to the way in which negative numbers were introduced. Faced with an otherwise unanswerable question, the Hindus merely defined –1 as the solution to the question, What is zero subtract one? It is easier to accept the concept of –1 only because we have experience of the analogous concept of “debt,” whereas we have nothing in the real world to underpin the concept of an imaginary number. The seventeenth-century German mathematician Gottfried Leibniz elegantly described the strange nature of the imaginary number: “The imaginary number is a fine and wonderful recourse of the divi...

With a few minor manipulations imaginary numbers turn out to be the ideal way to analyze the natural swinging motion of objects such as pendula. This motion, technically called a sinusoidal oscillation, is found throughout nature, and so imaginary numbers have become an integral part of many physical calculations. Nowadays electrical engineers conjure up i to analyze oscillating currents, and theoretical physicists calculate the consequences of oscillating quantum mechanical wave functions by summoning up the powers of imaginary numbers. Pure mathematicians have also exploited imaginary numbers, using them to find answers to previously impenetrable problems. Imaginary numbers literally add a new dimension to mathematics, and Euler hoped to exploit this extra degree of freedom to attack Fermat’s Last Theorem.

The proof for the case n = 4 also proves the cases n = 8, 12, 16, 20,….The reason is that any number that can be written as an 8th (or a 12th, 16th, 20th,…) power can also be rewritten as a 4th power. For instance, the number 256 is equal to 28, but it is also equal to 44. Therefore any proof that works for the 4th power will also work for the 8th power and for any other power that is a multiple of 4. Using the same principle, Euler’s proof for the case n = 3 automatically proves the cases n = 6, 9, 12, 15,…

all other numbers can be created by multiplying combinations of the prime numbers. This seems to lead to a remarkable breakthrough. To prove Fermat’s Last Theorem for all values of n, one merely has to prove it for the prime values of n. All other cases are merely multiples of the prime cases and would be proved implicitly. Intuitively this enormously simplifies the problem, because you can ignore those equations that involve a value of n that is not a prime number. The number of equations remaining is now vastly reduced. For example, for the values of n up to 20, there are only six values that need to be proved: If one can prove Fermat’s Last Theorem for just the prime values of n, then the theorem is proved for all values of n. If one considers all whole numbers, then it is obvious that there are infinitely many. If one considers just the prime numbers, which are only a small fraction of all the whole numbers, then surely the problem is much simpler?

Hilbert’s Hotel seems to suggest that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel—the infinity of even numbers can be matched up and compared with the infinity of all counting numbers. However, some infinities are indeed bigger than others. For example, any attempt to pair every rational number with every irrational number ends in failure, and in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers.

Maria Agnesi was born in Milan in 1718 and, like Hypatia, was the daughter of a mathematician. She was acknowledged to be one of the finest mathematicians in Europe, particularly famous for her treatises on the tangents to curves. In Italian, curves were called versiera, a word derived from the Latin vertere, “to turn,” but it was also an abbreviation for avversiera, or “wife of the Devil.” A curve studied by Agnesi (versiera Agnesi) was mistranslated into English as the “witch of Agnesi,” and in time the mathematician herself was referred to by the same title.

Germain’s contribution may have been forever wrongly attributed to the mysterious Monsieur Le Blanc were it not for the Emperor Napoleon. In 1806 Napoleon was invading Prussia and the French army was storming through one German city after another. Germain feared that the fate that befell Archimedes might also take the life of her other great hero, Gauss, so she sent a message to her friend General Joseph-Marie Pernety, who was in charge of the advancing forces. She asked him to guarantee Gauss’s safety, and as a result the general took special care of the German mathematician, explaining to him that he owed his life to Mademoiselle Sophie Germain. Gauss was grateful but surprised, for he had never heard of the mysterious lady. The game was up. In Germain’s next letter to Gauss she reluctantly revealed her true identity. Far from being angry at the deception, Gauss wrote back to her with delight: But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encou...

Unique factorization states that there is only one possible combination of primes that will multiply together to give any particular number.

However, if you examine the arrangement that was being sold by Loyd, in which the 14 and 15 were swapped, then the value of the disorder parameter is one, Dp =1, i.e., the only pair of tiles out of order are the 14 and 15. For Loyd’s arrangement the disorder parameter has an odd value! Yet we know that any arrangement derived from the correct arrangement has an even value for the disorder parameter. The conclusion is that Loyd’s arrangement cannot be derived from the correct arrangement, and conversely it is impossible to get from Loyd’s arrangement back to the correct one—Loyd’s $1,000 was safe. Loyd’s puzzle and the disorder parameter demonstrate the power of an invariant. Invariants provide mathematicians with an important strategy to prove that it is impossible to transform one object into another. For example, an area of current excitement concerns the study of knots, and naturally knot theorists are interested in trying to prove whether or not one knot can be transformed into another by twisting and looping but without cutting. In order to answer this question they attempt to find a property of the first knot that cannot be destroyed no matter how much twisting and looping occurs—a knot invariant. They then calculate the same property for the second knot. If the values are different then the conclusion is that it must be impossible to get from the first knot to the second.

G. H. Hardy tried to explain and justify his own career in a book entitled A Mathematician’s Apology: I will only say that if a chess problem is, in the crude sense, “useless,” then that is equally true of most of the best mathematics…I have never done anything “useful.” No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value. The desire for a solution to any mathematical problem is largely fired by curiosity, and the reward is the simple but enormous satisfaction derived from solving any riddle. The mathematician E. G. Titchmarsh once said: “It can be of no practical use to know that π is irrational, but if we can know, it surely would be intolerable not to know.”

This is a number beyond the imagination, and beyond any practical application. Hardy called Skewes’s number “the largest number that has ever served any definite purpose in mathematics.” He calculated that if one played chess with all the particles in the universe (1087), where a move meant simply interchanging any two particles, then the number of possible games was roughly Skewes’s number.

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. G. H. Hardy

In due course André Weil, one of the godfathers of twentieth-century number theory, was to adopt the conjecture and publicize it in the West. Weil investigated the idea of Shimura and Taniyama, and found even more solid evidence in favor of it. As a result, the hypothesis was often referred to as the Taniyama-Shimura-Weil conjecture, sometimes as the Taniyama-Weil conjecture, and occasionally as the Weil conjecture. In fact there has been much debate and controversy over the official naming of the conjecture. For those of you interested in combinatorics there are fifteen possible permutations given the three names involved, and it is quite probable that every one of those combinations has appeared in print over the years. However, I will refer to the conjecture by its original title, the Taniyama-Shimura conjecture.

Mathematicians studying elliptic equations might not be well versed in things modular, and conversely. Then along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds. Mathematicians love to build bridges.” The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s creations. Mathematics consists of islands of knowledge in a sea of ignorance. For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathematicians discuss risk and chance. There are dozens of such islands, each one with its own unique language, incomprehensible to the inhabitants of other islands. The language of geometry is quite different from the language of probability, and the slang of calculus is meaningless to those who speak only statistics. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time.

Gerhard Frey had come to the dramatic conclusion that the truth of Fermat’s Last Theorem would be an immediate consequence of the Taniyama-Shimura conjecture being proved. Frey claimed that if mathematicians could prove the Taniyama-Shimura conjecture then they would automatically prove Fermat’s Last Theorem. For the first time in a hundred years the world’s hardest mathematical problem looked vulnerable. According to Frey, proving the Taniyama-Shimura conjecture was the only hurdle to proving Fermat’s Last Theorem.

An expert problem solver must be endowed with two incompatible qualities—a restless imagination and a patient pertinacity. Howard W. Eves

Galois was desperate to attend the Polytechnique, not just because of its academic excellence but also because of its reputation for being a center for republican activism. One year later he reapplied and once again his logical leaps in the oral examination only served to confuse his examiner, Monsieur Dinet. Sensing that he was about to be failed for a second time and frustrated that his brilliance was not being recognized, Galois lost his temper and threw a blackboard eraser at Dinet, scoring a direct hit. Galois was never to return to the hallowed halls of the Polytechnique.

In July of 1829 a new Jesuit priest arrived in the village of Bourg-le-Reine, where Galois’s father was still mayor. The priest took exception to the mayor’s republican sympathies and began a campaign to oust him from office by spreading rumors aimed at discrediting him. In particular the scheming priest exploited Nicolas-Gabriel Galois’s reputation for composing clever rhymes. He wrote a series of vulgar verses ridiculing members of the community and signed them with the mayor’s name. The elder Galois could not survive the shame and the embarrassment that resulted and committed suicide.

As the coffin was being lowered into the grave, a scuffle broke out between the Jesuit priest, who was conducting the service, and supporters of the mayor, who realized that there had been a plot to undermine him. The priest suffered a gash to the head, the scuffle turned into a riot, and the coffin was left to drop unceremoniously into its grave. Watching the French establishment humiliate and destroy his father served only to consolidate Galois’s fervent support for the republican cause.

Last year before March 1st, Monsieur Galois gave to the secretary of the Institute a memoir on the solution of numerical equations. This memoir should have been entered in the competition for the Grand Prize in Mathematics. It deserved the prize, for it could resolve some difficulties that Lagrange had failed to do. Monsieur Cauchy had conferred the highest praise on the author about this subject. And what happened? The memoir is lost and the prize is given without the participation of the young savant. Le Globe, 1831 Galois felt that his memoir had been deliberately lost by a politically biased Academy, a belief that was reinforced a year later when the Academy rejected his next manuscript, claiming that “his argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor.” He decided that there was a conspiracy to exclude him from the mathematical community, and as a result he neglected his research in favor of fighting for the republican cause.

A week later a sniper in a garret opposite the prison fired a shot into a cell, wounding the man next to Galois. Galois was convinced that the bullet was intended for himself and that there was a government plot to assassinate him. The fear of political persecution terrorized him, and the isolation from his friends and family and rejection of his mathematical ideas plunged him into a state of depression. In a bout of drunken delirium he tried to stab himself to death, but his fellow republicans managed to restrain and disarm him.

Liouville reflected on why the young mathematician had been rejected by his seniors and how his own efforts had resurrected Galois: As Descartes said, “When transcendental questions are under discussion be transcendentally clear.” Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them ardent, active; he could profit by their advice. But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits… My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem.

No doubt inspired by all the media attention, a new piece of graffiti found its way onto New York’s Eighth Street subway station: xn + yn = zn: no solutions I have discovered a truly remarkable proof of this, but I can’t write it now because my train is coming.

The journalists covering the story tended to concentrate on Fermat and mentioned Taniyama-Shimura only in passing, if at all. Shimura, a modest and gentle man, was not unduly bothered by the lack of attention given to his role in the proof of Fermat’s Last Theorem, but he was concerned that he and Taniyama had been relegated from being nouns to adjectives. “It is very curious that people write about the Taniyama-Shimura conjecture, but nobody writes about Taniyama and Shimura.”

A problem worthy of attack Proves its worth by fighting back. Piet Hein