Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem

Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem

4.1 11
by Simon Singh

xn + yn = zn, where n represents 3, 4, 5, ...no solution

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

With these words, the seventeenth-century French mathematician Pierre de…  See more details below


xn + yn = zn, where n represents 3, 4, 5, ...no solution

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

With these words, the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations. What came to be known as Fermat's Last Theorem looked simple; proving it, however, became the Holy Grail of mathematics, baffling its finest minds for more than 350 years. In Fermat's Enigma—based on the author's award-winning documentary film, which aired on PBS's "Nova"—Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it. Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics.

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Editorial Reviews

Wall Street Journal
It is hard to imagine a more gripping account of . . .this centuries-long drama of ingenious failures, crushed hopes, fatal duels, and suicides
Boston Sunday Globe
Though Singh may not expect us to bring too much algebra to the table, he does expect us to appreciate a good detective story.
Christian Science Monitor
The amazing achievement of Singh's book is that it actually makes the logic of the modern proof understandable to the nonspecialist. . .More important, Signh shows why it is significant that this problem should have been solved.
Publishers Weekly - Publisher's Weekly
The 17th-century French amateur mathematician and all-around Renaissance man Pierre de Fermat posed what seems to be a simple mathematical theorem: you cannot find three numbers such that xn + yn = zn, where n is greater than 2. Fermat scribbled in the margin of a book that he had found a 'truly marvelous' proof, but he seemingly never bothered to write it out. Mathematicians sought this mathematical Holy Grail for over 300 years, many doubtful that it even existed, some killing themselves after failed pursuits, until English-born Princeton professor Andrew Wiles finally proved what came to be known as 'Fermat's Last Theorem' in 1994, and became an overnight celebrity.Much like a mathematician constructing a proof, Singh, a BBC science journalist with a Ph.D in particle physics, clearly explains various characteristics of numbers and then pulls them together to show how Wiles derived his complex solution. The history of mathematics comes alive even for those who dread balancing their checkbooks, and anyone interested in the creative process will appreciate Singh's insights into how a mathematician tackles such a monumental problem. Wiles may have proven Fermat's theorem, but an enigma remains: did Fermat really have a proof using the much less elaborate knowledge of his day, and was it correct? 'The Riddler' continues to taunt mathematicians.
School Library Journal
The riveting story of a mathematical problem that sprang from the study of the Pythagorean theorem developed in ancient Greece. The book follows mathematicians and scientists throughout history as they searched for new mathematical truths. In the 17th century, a French judicial assistant and amateur mathematician, Pierre De Fermat, produced many brilliant ideas in the field of number theory. The Greeks were aware of many whole number solutions to the Pythagorean theorem, where the sum of two perfect squares is a perfect square. Fermat stated that no whole number solutions exist if higher powers replace the squares in this equation. He left a message in the margin of a notebook that he had a proof, but that there was insufficient space there to write it down. His note was found posthumously, but the solution remained a mystery for 350 years. Finally, after working in isolation for eight years, Andrew Wiles, a young British mathematician at Princeton University, published a proof in 1995. Although this famous question has been resolved, many more remain unsolved, and new problems continually arise to challenge modern minds. This vivid account is fascinating reading for anyone interested in mathematics, its history, and the passionate quest for solutions to unsolved riddles. The book includes 19 black-and-white photos of mathematicians and occasional sketches of ancient mathematicians as well as diagrams of formulas. The illustrations help to humanize the subject and add to the readability. -- Penny Stevens, Centreville Regional Library, Centreville, Virginia
Sir Penrose
An excellent account of one of the most dramatic and moving events of the century. -- New York Times Book Review
An excellent account of one of the most dramatic and movine events of the century. -- The New York Times Book Review
Christian Sci. Monitor
The amazing achievement of Singh's book is that it actually makes the logic of the modern proof understandable to the nonspecialist. . .More important, Signh shows why it is significant that this problem should have been solved.
Kirkus Reviews
The proof of Fermat's Last Theorem has been called the mathematical event of the century; this popular account puts the discovery in perspective for non-mathematicians. As one of the producers of the BBC 'Horizons' show on how the 300-year-old puzzle was solved, Singh had ample opportunity to interview Andrew Wiles, the Princeton professor who made the historic breakthrough. As a schoolboy in England, Wiles stumbled across a popular account of Fermat's puzzle: the assertion that no pair of numbers raised to a power higher than two can add up to a third number raised to the same power. Singh traces the roots of the problem in ancient geometry, from the school of Pythagoras (whose famous theorem is clearly its inspiration) up to the flowering of mathematics in the Renaissance, when Fermat, a French judge who dabbled in number theory, stated the problem and claimed to have found a proof of it. Generations of the finest mathematicians failed to corroborate his claim. Singh gives a colorful and generally easy-to-follow summary of much of the mathematical theory that was generated in attempts to prove Fermat's conjecture. Finally, in the 1950s, two Japanese mathematicians came up with a conjecture concerning elliptical equations that, at the time, seemed to have nothing to do with Fermat's problem. But it was the Taniyama-Shimuru conjecture that gave Wiles the opening to solve the problem after working in isolation for seven years. He announced his proof at a famous mathematical congress in Cambridge, England—a truly great moment in mathematical history. Then a flaw in the proof presented itself—and Wiles went back to work for over a year to patch it up. Finally he succeeded, andthe greatest problem in mathematical history was laid to rest. A good overview of one of the great intellectual puzzles of modern history.

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The Last Problem

In 1963, when he was ten years old, Andrew Wiles was already fascinated by mathematics. "I loved doing the problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found I discovered in my local library."

One day, while wandering home from school, young Wiles decided to visit the library in Milton Road. It was rather small, but it had a generous collection of puzzle books, and this is what often caught Andrew's attention. These books were packed with all sorts of scientific conundrums and mathematical riddles, and for each question the solution would be conveniently laid out somewhere in the final few pages. But this time Andrew was drawn to a book with only one problem, and no solution.

The book was The Last Problem by Eric Temple Bell. It gave the history of a mathematical problem that has its roots in ancient Greece, but that reached full maturity only in the seventeenth century when the French mathematician Pierre de Fermat inadvertently set it as a challenge for the rest of the world. One great mathematician after another had been humbled by Fermat's legacy, and for three hundred years nobody had been able to solve it.

Thirty years after first reading Bell's account, Wiles could remember how he felt the moment he was introduced to Fermat's Last Theorem: "It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem that I, a ten-year-old, could understand and I knew from that moment that I would never let it go. I had to solve it."

Usually half the difficulty in a mathematics problem is understanding the question, but inthis case it was straightforward--prove that there are no whole number solutions for this equation:

xn + yn = zn for n greater than 2.

The problem has a simple and familiar look to it because it is based on the one piece of mathematics that everyone can remember--Pythagoras's theorem:

In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

Or: x2+ y2= z2

Pythagoras's theorem has been scorched into millions if not billions of human brains. It is the fundamental theorem that every innocent schoolchild is forced to learn. But despite the fact that it can be understood by a ten-year-old, Pythagoras's creation was the inspiration for a problem that had thwarted the greatest mathematical minds of history.

In the sixth century B.C., Pythagoras of Samos was one of the most influential and yet mysterious figures in mathematics. Because there are no firsthand accounts of his life and work, he is shrouded in myth and legend, making it difficult for historians to separate fact from fiction. What seems certain is that Pythagoras developed the idea of numerical logic and was responsible for the first golden age of mathematics. Thanks to his genius numbers were no longer merely used to count and calculate, but were appreciated in their own right. He studied the properties of particular numbers, the relationships between them, and the patterns they formed. He realized that numbers exist independently of the tangible world and therefore their study was untainted by the inaccuracies of perception. This meant he could discover truths that were independent of opinion or prejudice and that were more absolute than any previous knowledge.

He gained his mathematical skills on his travels throughout the ancient world. Some tales would have us believe that Pythagoras traveled as far as India and Britain, but what is more certain is that he gathered many mathematical techniques from the Egyptians and Babylonians. Both these ancient peoples had gone beyond the limits of simple counting and were capable of performing complex calculations that enabled them to create sophisticated accounting systems and construct elaborate buildings. Indeed they saw mathematics as merely a tool for solving practical problems; the motivation behind discovering some of the basic rules of geometry was to allow reconstruction of field boundaries that were lost in the annual flooding of the Nile. The word itself, geometry, means "to measure the earth."

Pythagoras observed that the Egyptians and Babylonians conducted each calculation in the form of a recipe that could be followed blindly. The recipes, which would have been passed down through the generations, always gave the correct answer and so nobody bothered to question them or explore the logic underlying the equations. What was important for these civilizations was that a calculation worked--why it worked was irrelevant.

After twenty years of travel Pythagoras had assimilated all the mathematical rules in the known world. He set sail for his home island of Samos in the Aegean Sea with the intention of founding a school devoted to the study of philosophy and, in particular, concerned with research into his newly acquired mathematical rules. He wanted to understand numbers, not merely exploit them. He hoped to find a plentiful supply of free-thinking students who could help him develop radical new philosophies, but during his absence the tyrant Polycrates had turned the once liberal Samos into an intolerant and conservative society. Polycrates invited Pythagoras to join his court, but the philosopher realized that this was only a maneuver aimed at silencing him and therefore declined the honor. Instead he left the city in favor of a cave in a remote part of the island, where he could contemplate without fear of persecution.

Pythagoras did not relish his isolation and eventually resorted to bribing a young boy to be his first pupil. The identity of the boy is uncertain, but some historians have suggested that his name was also Pythagoras, and that the student would later gain fame as the first person to suggest that athletes should eat meat to improve their physiques. Pythagoras, the teacher, paid his student three oboli for each lesson he attended and noticed that as the weeks passed the boy's initial reluctance to learn was transformed into an enthusiasm for knowledge. To test his pupil Pythagoras pretended that he could no longer afford to pay the student and that the lessons would have to stop, at which point the boy offered to pay for his education rather than have it ended. The pupil had become a disciple. Unfortunately this was Pythagoras's only conversion on Samos. He did temporarily establish a school, known as the Semicircle of Pythagoras, but his views on social reform were unacceptable and the philosopher was forced to flee the colony with his mother and his one and only disciple.

Pythagoras departed for southern Italy, which was then a part of Magna Graecia, and settled in Croton, where he was fortunate in finding the ideal patron in Milo, the wealthiest man in Croton and one of the strongest men in history. Although Pythagoras's reputation as the sage of Samos was already spreading across Greece, Milo's fame was even greater. Milo was a man of Herculean proportions who had been champion of the Olympic and Pythian Games a record twelve times. In addition to his athleticism Milo also appreciated and studied philosophy and mathematics. He set aside part of his house and provided Pythagoras with enough room to establish a school. So it was that the most creative mind and the most powerfu1 body formed a partnership.

Secure in his new home, Pythagoras founded the Pythagorean Brotherhood--a band of six hundred followers who were capable not only of understanding his teachings, but who could add to them by creating new ideas and proofs. Upon entering the Brotherhood each follower had to donate all his worldly possessions to a common fund, and should anybody ever leave he would receive twice the amount he had originally donated and a tombstone would be erected in his memory. The Brotherhood was an egalitarian school and included several sisters. Pythagoras's favorite student was Milo's own daughter, the beautiful Theano, and, despite the difference in their ages, they eventually married.

Soon after founding the Brotherhood, Pythagoras coined the word philosopher, and in so doing defined the aims of his school. While attending the Olympic Games, Leon, Prince of Phlius, asked Pythagoras how he would describe himself. Pythagoras replied, "I am a philosopher," but Leon had not heard the word before and asked him to explain.

Life, Prince Leon, may well be compared with these public Games for in the vast crowd assembled here some are attracted by the acquisition of gain, others are led on by the hopes and ambitions of fame and glory. But among them there are a few who have come to observe and to understand all that passes here.

It is the same with life. 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.

Although many were aware of Pythagoras's aspirations, nobody outside of the Brotherhood knew the details or extent of his success. Each member of the school was forced to swear an oath never to reveal to the outside world any of their mathematical discoveries. Even after Pythagoras's death a member of the Brotherhood was drowned for breaking his oath--he publicly announced the discovery of a new regular solid, the dodecahedron, constructed from twelve regular pentagons. The highly secretive nature of the Pythagorean Brotherhood is part of the reason that myths have developed surrounding the strange rituals that they might have practiced, and similarly this is why there are so few reliable accounts of their mathematical achievements.

What is known for certain is that Pythagoras established an ethos that changed the course of mathematics. The Brotherhood was effectively a religious community, and one of the idols they worshiped was Number. By understanding the relationships between numbers, they believed that they could uncover the spiritual secrets of the universe and bring themselves closer to the gods. In particular tbe Brotherhood focused its attention on the study of counting numbers (1, 2, 3, ...) and fractions. Counting numbers are sometimes called whole numbers, and, together with fractions (ratios between whole numbers), they are technically referred to as rational numbers. Among the infinity of numbers, the Brotherhood looked for those with special significance, and some of the most special were the so-called "perfect" numbers.

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.

As well as having mathematical significance for the Brotherhood, the perfection of 6 and 28 was acknowledged by other cultures who observed that the moon orbits the earth every 28 days and who declared that God created the world in 6 days. In The City of God, St. Augustine argues that although God could have created the world in an instant he decided to take six days in order to reflect the universe's perfection. St. Augustine observed that 6 was not perfect because God chose it, but rather that the perfection was inherent in the nature of the number: "6 is a number perfect in itself, and not because God created all things in six days; rather the inverse is true; God created all things in six days because this number is perfect. And it would remain perfect even if the work of the six days did not exist."

As the counting numbers get bigger the perfect numbers become harder to find. The third perfect number is 496, the fourth is 8,128, the fifth is 33,550,336, and the sixth is 8,589,869,056. As well as being the sum of their divisors, Pythagoras noted that all perfect numbers exhibit several other elegant properties. For example, perfect numbers are always the sum of a series of consecutive counting numbers. So we have

6 =1+2+3,
28 =1+2+3+4+5+6+7,
496=1+2+3+4+5+6+7+8+9+ . . . +30+31,
8,128 =1+2+3+4+5+6+7+8+9+ . . . +126+127.

Pythagoras was entertained by perfect numbers, but he was not satisfied with merely collecting these special numbers; instead he desired to discover their deeper significance. One of his insights was that perfection was closely linked to "twoness." The numbers 4 (2x2), 8 (2 x2 x 2), 16 (2x 2 x 2x 2), etc., are known as powers of 2, and can be written as 2n, where the n represents the number of 2's multiplied together. All these powers of 2 only just fail to be perfect, because the sum of their divisors always adds up to one less than the number itself.

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Meet the Author

Simon Singh received his Ph.D. in physics from the University of Cambridge. A BBC producer, he directed and coproduced an award-winning documentary film on Fermat's Last Theorem that aired on PBS's "Nova" series. He is also the author of the bestselling The Code Book:  The Science of Secrecy from Ancient Egypt to Quantum Cryptography.  He lives in London, England.

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Fermat's Enigma 5 out of 5 based on 0 ratings. 1 reviews.
Guest More than 1 year ago
Reads like a great detective story. The fact that its true makes it even more enjoyable. Try it, you'll like it -- really!!