A Fermat belief leaves no margin for doubt

Andrew Wiles won a $-670,000 Abel Prize last week for proving Fermat's Last Theorem. The real reward, though, was scaling one of the world’s greatest mathematical adventures on the back of sheer tenacity and the shoulders of giants.

March 24, 2016 03:55 am | Updated October 18, 2016 02:02 pm IST

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A few days after I got married, as my wife and I were retiring for the night, she asked me if I needed to keep the lights on so that I could read a book. I said yes, and I opened one I had read thrice already.

“You are going to read that ?” she asked. She then looked up to the ceiling, folded her hands, and wondered in a jocular manner, “O dear lord, what have I gotten into?”

The book I was reading, the one that got my wife to exclaim thus, was Simon Singh’s Fermat’s Last Theorem , a riveting account of what is easily one of the greatest mathematical adventures of all time: the incredibly humbling story of Andrew Wiles, and how he solved one of the toughest, most intriguing mathematical conundrums in history. I later also saw Dr. Singh’s extraordinary documentary on the same topic, broadcast on the BBC as part of the ‘ Horizon ’ series.

Since then, I have read that book at least 11 times more, and watched the documentary innumerable times. It is better than the Bourne series of spy thrillers; it’s that good.

Perhaps because of the book, or perhaps because he solved something that all of us maths students had naively attempted and miserably failed to while in school or college, Professor Wiles became one of my best-loved mathematicians.

It was so simple a question that almost everybody thought he or she could solve it, but as it was later noted, Fermat’s Last Theorem could not have been proved in the 17th, the 18th or the 19th centuries because the knowledge needed to solve it was provided only by the works of mathematicians of the 20th.

Wiles’ was an extraordinary feat of solo work, but also one that depended on so many to provide the groundwork. As Sir Isaac Newton wrote in one letter, “We (the moderns) are like dwarves perched on the shoulders of giants (the ancients), and thus we are able to see more and see farther than them... we are carried aloft and elevated by the magnitude of the giants.”

For seven years, Wiles was at it, at Cambridge, to conquer the elusive proof. He relied on the works of not only Fermat, but also Evariste Galois, Sophie Germain, Leonhard Euler, Ken Ribet, Gerhard Frey, Goro Shimura, Yutaka Taniyama, Jean-Pierre Serre, Matthias Flach, Victor Kolyvagin, Kenkichi Iwasawa, and several others whose work he not only quoted but also used as a platform to prove the theorem.

When it was proved, in June 1993, it was on the front pages of The New York Times and most other newspapers around the world. A few months later, though, it turned out that the proof had a fatal flaw. Wiles’ friend and referee Nick Katz found that portions of the proof were incorrect, and his work collapsed, like a house of cards. The mathematics world was stunned, almost speechless.

Instead of accepting defeat and going back to a career of anonymity, Wiles went back to his room, determined to find a solution. A year later, in September 1994, Wiles was struck by a most singular idea that could overturn the flaw. It was an Archimedes moment, and while the shy Wiles would never have run naked around the Princeton campus shouting eureka, he did stare at the paper for 20 minutes, go out for a walk, then come back to see whether it was still there. It was still there. He says he slept the night, and checked back again the next morning. It was still there.

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

... wrote Fermat in 1637

He then went to his wife and said, “I think I have found it.” She thought he had found their kid’s toy.

Wiles’ story is not just of getting the proof, but also of not giving up. To bring up a cricket analogy, it was when the world discovered that Mutthiah Muralitharan had a fatal flaw in his bowling action that he went back to an academy, got it corrected, and came back to become the world’s highest wicket-taker in both Tests and One-Day Internationals.

So, last week when Wiles was announced as the winner of the Abel Prize, which is often referred to as the mathematics Nobel, I was over the moon. As, I am certain, were his fans all over the world.

There are other 20th-century mathematicians who are extraordinary stories in themselves: Grigori Perelman, Yitang Zhang, Yutaka Taniyama (and his friend Goro Shimura), John Nash, Srinivasa Ramanujan, Alexander Grothendieck, Kurt Godel, Paul Erdos, John von Neumann, the list is endless. If you want younger names, add Terence Tao and Manjul Bhargava to that group. But Wiles’ proof is interesting because it is about something we see right from our school days; the Pythagoras theorem about right-angled triangles. Wiles discovered the theorem as a 10-year-old, little knowing its significance to his life 30 years later.

As any schoolchild would know, for a right-angled triangle with sides a, b, and c, “a 2 + b 2 = c 2 ” (where c is the hypotenuse). That is to say, in a right-angled triangle, the sum of the squares of the two smaller sides is equal to the square of the hypotenuse.

This is called the Pythagoras theorem, but there is no documentary evidence that Pythagoras postulated it. Indian mathematicians were using this property (as were Babylonians) much before the Greek philosopher did. It is possible that he wrote the first proof (and, in mathematics, nothing is accepted without proof), or maybe his students did. No one knows.

In fact, in his book on the most influential mathematicians in history, Men of Mathematics , mathematician and science fiction writer Eric Temple Bell does not even mention Pythagoras.

Source: The Hindu Archives

>Pierre de Fermat (1601~1665) was a French mathematician who discovered the fundamental principle of analytic geometry. His methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the differential calculus. Through his correspondence with Blaise Pascal he was a co-founder of the theory of probability.

Not the first scholar in history to be thwarted by the size of the margin in his textbook.

It was in this book, in 1989, that I was introduced to Pierre de Fermat, the man who postulated his eponymous theorem, which says that “a n + b n = c n ” is not true for any 'n' value greater than 2. He even claimed that he had a “truly marvellous proof” for this, but that the margins of the book were too small to fit it in (the book he was referring to was Arithmetica by the legendary Greek mathematician Diophantus, better known as the father of algebra).

Fermat wrote in 1637: “If an integer n is greater than 2, then a n + b n = c n has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”

Born in the first decade of the 17th century (there is no consensus on whether he was born in 1601 or 1607), Fermat led an interesting life. He belonged to a rich family; so rich that he could buy a councillor’s seat in his hometown of Toulouse, in France. This entitled him to practise law. While he was a practising lawyer (he never left the profession till his death), Fermat would also spend a lot of time thinking about mathematics, and had a special interest in number theory and geometry. Dr. Bell calls him “the prince of amateurs”.

Along with Blaise Pascal, another French intellectual giant, Fermat invented the mathematics of probability, the basis for the science of statistics. There is no academic record of his learning mathematics, so it must have been mere amusement, a sort of recreation for him.

Throughout his distinguished career as a jurist, Fermat never faltered even once, and was known as a just and fair man. In his pastime, though, he was a prolific mathematical genius who made vital contributions to analytic geometry, adequality, number theory, and even optics (Fermat’s Principle of Least Time is about how a ray of light travels between two points). But his greatest contribution to mathematical lore is the postulation of Fermat’s enigma, also called the Last Theorem.

For more than 300 years, hundreds — no, thousands — of mathematicians have tried to solve the equation. Even Leonhard Euler, whose contributions to mathematics are second to none, tried to solve it, but he could prove it only up to n = 4. Well, you could, theoretically, prove it for n = 5, n = 6, and so on, and so forth, all the way up to infinity. But mathematics needs, indeed demands, universal proof. In any case, no one knows what infinity is, so there is no point in proving Fermat integer by integer. It will take till the end of time, and you’d still be nowhere close to solving it.

In the 1970s, mathematicians tried to use the most complex computers in use then to prove the equation. They could do so for prime numbers ranging from 3 all way up to 4 million. But what next? There is no limit to primes, so that exercise could go on forever as well. Clearly, computers were not the way to do it. It was like striking a rhino with a lead pencil. No matter how many times you hit the animal, it won’t make any difference whatsoever.

It was not that mathematicians did not try. Even businessmen interested in mathematics announced grand monetary prizes to anyone who solves it. Simon Singh notes in his book that the French Academy of Sciences announced a cash prize of 3,000 francs to the person who could solve it. This was in 1816. Later, in 1883, the Academy of Brussels (Belgium) announced its own prize. In 1908, writes Singh, a German millionaire Paul Wolfskehl pledged a huge sum to anyone who could solve it. Naturally, all kinds of proofs came in. None of them was right.

None of them, that is, until May 1995, when Prof. Wiles’ final proof was published. He collected his booty in 1997 (two years after it was proved, as per the condition set by Wolfskehl). By then, the Wolfskehl prize money was worth $50,000. In comparison, the Abel Prize he won last week is worth 600,000 euros ($670,000).

Prof. Wiles began to work on the theorem in 1986. In 1984, a huge breakthrough happened when Gerhard Frey, a German mathematician who blazed several trails in number theory, set certain conditions for Fermat’s Last Theorem to be proved. One of them — the epsilon conjecture — was proved in 1986 by Ken Ribet (who, in turn, depended on Frey’s seminal work on elliptic curves), and this formed the basis of Wiles’ proof.

The biggest challenge for Wiles, really, was to solve what was then called the >Taniyama-Shimura-Weil conjecture (that all elliptic curves are modular). It was later called the Modularity Theorem. Taniyama, a troubled soul, committed suicide in 1958, before anyone could prove the conjecture. “I was puzzled (by his suicide)… I was unable to make sense of it,” says Prof. Shimura in Singh’s documentary. Nevertheless, the work by the two Japanese mathematicians would prove to be pivotal to Wiles 37 years later.

Wiles’ theorem, in essence, proves that all elliptic curves are modular, fulfilling Frey’s second condition to prove Fermat. It was in September 1994 that Wiles had finally got the correct proof, and by October, it was validated by his senior colleagues. It was only then that he submitted two manuscripts (written with his former student Richard Taylor) to Annals of Mathematics . Both were published in May 1995. 358 years after Fermat had posed the problem, Andrew Wiles, the son a divinity professor, had proved it.

Perhaps it was just as well. Because (if you are a believer), mathematics comes closest to understanding the mind of god. Or as theoretical physicist Michio Kaku of City University, New York, says, “Math is the mind of god.”

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