Fermat's Last Theorem - Simon Singh

Can you picture yourself working for seven years in solitude and secrecy on a problem that has stumped the world's best mathematicians for centuries? Imagine, if you can, that your work is crowned with success, and that overnight you become a worldwide celebrity. Then, after a few months, your peers point out a hole in your argument. Back to your solitary pondering, only this time you cannot escape the glare of publicity!

This nightmare predicament was the reality Andrew Wiles found himself in for a whole year following his initial announcement of success. Miraculously, just as he was ready to finally admit defeat, he had a flash of insight on how he could overcome the remaining obstacle and complete his proof.

Pierre de Fermat (1601-1665) was a legendary French mathematician, but he would hardly still be remembered by laymen, had it not been for his famous scribbling in 1637. In the margin of his copy of an ancient Greek text (Diophant's Arithmetica), where he stated his so-called last theorem in latin, he added: "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.)

The theorem is deceptively simple: No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. If n = 2, the formula becomes Pythagoras' theorem for the sides of a right-angled triangle, which has an infinite number of solutions, e.g. 3*3 + 4*4 = 5*5 or 5*5 + 12*12 = 13*13.

For more than 350 years, all the best mathematicians (and countless amateurs) tried in vain to re-discover the proof that Fermat claimed to have found. Over time doubts have grown, tempered by the knowledge that Fermat was a mathematical genius, and that he had the habit of offering his discoveries as challenges to his colleagues. Wiles' proof could not have been found by Fermat, but we cannot dismiss Fermat's claim with complete certainty.

Simon Singh's spellbinding book describes Wiles' lonely struggle to find a proof, his false success, his setback, and his final victory against all odds. Per aspera ad astra! Along the way we also learn a lot about the cultural history of mathematics.

You do not need a background in mathematics to enjoy the book. Although I have a degree in engineering physics, I must admit that I do not have the faintest idea of what a statement such as "There is a close correspondence between elliptic curves and modular forms" means. And I had never heard of "the Taniyama-Shimura conjecture" before reading this book. Sounds like something right out of Star Trek, doesn't it? In fact, probably no more than a hundred mathematicians in the world, or so, are able to follow Wiles' reasoning in detail. Item 6 in the list below will show you why! Yet, Singh succeeds in conveying the essence of the long mathematical journey which finally led to the proof.

I find the account of Wiles' personal struggle at least as fascinating as the puzzle itself. What did he actually do during those seven solitary years, except sit at his desk, stare at the formula, and chew on his pencil? And why the secrecy? Most of us enjoy working in groups or in tandem. We need someone to encourage us, to exchange ideas, to ask questions, to make us see things from a new angle.

Andrew Wiles (b. 1953).
Credit: D.Applewhite, Princeton University

It appears that Wiles is an intensely private person. Born in Cambridge, he had been dreaming of finding a proof of Fermat's Theorem since he was 10. He became a professor at Princeton University in the United states in the 1980s. He got married shortly after embarking on his intellectual odyssey and told his wife about his obsession during their honeymoon. They had two children, and his family gave him all the comfort and recreation he needed. He knew that his colleagues would not leave him alone if he told them the true nature of his research, and he felt that he needed to work with total concentration, without any distraction. During the latter stages of his work, he must also have feared that someone might leapfrog him and attain victory by building on his ideas. Shortly before publication he did confide in a trusted colleague who helped him examine the proof.

As for the nature of his work, he has likened it to stumbling around in a dark room in an unfamiliar mansion. Gradually you get some sense of where the furniture is, and how various objects feel to the touch. After six months you find the light switch and everything becomes clear! Then you move on to the next darkened room...

One can only try to imagine the emotional rollercoaster Wiles must have been on in 1993-94. First a series of lectures at Cambridge University before an elite audience in an electrified atmosphere, culminating in Wiles' simply writing Fermat's formula on the blackboard and saying: "I think I'll stop there." Then his e-mail correspondence with the examiners of his manuscript which made him realize his error. Then his year-long desperate attempts to complete his proof. He had given himself until the end of September 1994 before he would finally give up. Then, on September 19th, he had what he describes as a fantastic revelation that showed him how to conquer the remaining obstacle. He stared at his finding for twenty minutes in disbelief. During the rest of the day he kept returning to see if it was still there. The next morning he re-checked his result and after a few hours went to tell his wife. His emotions are still very much in evidence in Singh's and Lynch's film (see the list below).

I am reminded of Harry Martinson's Aniara, where a mathematical discovery is described as the birth of a child (in my imperfect and rough translation):

A discovery never anticipated has been made
by Isagel, the female pilot.
One morning she sat quietly in the Gopta room
occupied with Jender curves.

Then she cried out and asked me over
to the Jender table where in a flash she had caught,
in a provisionally closed form, her discovery.
She shouted with joy, pressing to her heart
her inspiration, kicking lively,
which borne by her had propitiously been begotten
in love of the Law of Large Numbers.

How certain can we be that there are no additional holes in Wiles' proof? The probability should be rather low, now that it has been out for 15 years, and has attracted great interest from the most eminent experts in the field. Fermat's Theorem is much more than a historical curiosity, because whole branches of mathematics have sprouted from investigations related to the Theorem. Regrettably, perhaps, mathematics has advanced so far that only the experts can express an opinion about a proof with any claim to credibility. Gone forever are the simple days when mathematical proofs could be demonstrated to schoolchildren, e.g. that the square root of two cannot be a rational number, or that there must be an infinite number of primes.

In the same vein, it may be even more difficult to pass judgment on mathematical theorems proven with the assistance of computers. (Wiles did not use one.) For instance, there is a theorem which says that any map, no matter how complex, requires no more than four colors to ensure that regions with a common border are assigned different colors. To a layman, it is surprising that this simple theorem was not proved to be true until 1976, using a computer to check 1936 different configurations. Checking the proof "by hand" seems impractical. Instead the proof must involve checking the programming of the computer itself. (This aspect is discussed in ref. 7 below.) And before long, computers may well turn into more than mere assistants in finding mathematical proofs, with staggering implications!

Further viewing and reading

1. Fermat's Last Theorem (1996). A documentary film. 45 min. "Simon Singh and John Lynch's film tells the enthralling and emotional story of Andrew Wiles."

2. An interview with Andrew Wiles.

3. Wikipedia article on Fermat, with many links.

4. Wikipedia article on Fermat's Last Theorem, with many links.

5. Wikipedia overview of Wiles' proof of Fermat's Last Theorem, with many links.

6. Wiles' 1995 article in Annals of Mathematics: "Modular Elliptic Curves and Fermat's Last Theorem". Highly technical. Unreadable to most of us, but it has a certain charm... Takes a minute or so to load.

7. 2005 Web article by Wim H. Hesselink.

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    Last edited or checked June 5, 2013. Broken link fixed March 16, 2018.

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