Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem 
by Amir Aczel.
Viking, 147 pp., £9.99, May 1997, 0 670 87638 0
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Aheroic story: Andrew Wiles, a Cambridge mathematician living in the United States, emerges after seven years of self-incarceration and paranoid secrecy from his Princeton attic, clutching two hundred pages of hieroglyphics. He is triumphant. He has cracked the most famous problem in number theory: Fermat’s Last Theorem, which has eluded some of the finest efforts of mathematicians for over three hundred years. He is beside himself with anticipated glory, but holds off, maintaining secrecy until the right moment. Serendipitously, a conference on exactly the branch of number theory covering his work is to take place in Cambridge. Choosing an unrevealing title, he lectures on three consecutive days. In the final minute of the final hour, he is able to utter the magic claim that the puzzle of Fermat’s celebrated theorem is now solved. Shocked silence and then a standing ovation from his astonished international colleagues. In the ruthlessly competitive world of research mathematics, Wiles has pulled off one of the most dramatic successes anyone could hope for.

As a sixth-former, I tried to prove Fermat’s Last Theorem. A little later, at university, I tried to prove Goldbach’s conjecture, disprove the twin-prime conjecture, generalise Riemann’s zeta function, settle the four-colour problem and solve various other enigmas, not least that of uncountable infinity, locked inside Cantor’s Continuum Hypothesis. I failed in all of these, but not being able to crack such puzzles only confirmed their attraction. Tackling them was self-confirmation not combat, making sure they were real, that I was kindred to all the other great names who’d tried and failed, that mathematics was as illustrious and difficult as it was said to be. Over the next ten years, working on my PhD and then teaching and researching, I spent hundreds of hours in silent rapport with the symbols that live in the depths. Sometimes, a whole day would be consumed as I paced and sat and scribbled and stared at my symbols, head filled with pictures, shapes, movements, patterns, connections on the edge of perception and language that I’d be trying to coax or bully into some stable, and ultimately logical/communicable, form. In those days, I found the meditations and imaginary journeys intensely satisfying.

This is presumably the kind of experience Andrew Wiles was having up there in his attic. What exactly was he doing? Anybody who has ever been fascinated, however briefly, by the extraordinary patterns numbers make has some idea. Number theory has been called the most ‘beautiful and treacherous’ of all the branches of mathematics: it is as easy to ask a profoundly difficult question as it is to demonstrate an elegant regularity. For example, one might observe that each even number is a sum of two primes (6 = 3 + 3, 8 = 3 + 5, 10 = 3 +7, 20 = 7 +13 and so on) and guess that all even numbers have this property. Nobody has found an even number which doesn’t have it, and nobody has proved that all numbers must have it; the hypothesis that they do, Goldbach’s conjecture, remains unresolved more than two centuries after its first appearance. Simple to state and understand, and seemingly impossible to settle, the example could be multiplied many times over.

Basically, number theory is theoretical arithmetic: it studies such things as the behaviour of prime numbers or integer solutions to equations. The pursuit is as old as mathematics: the Babylonians listed several sets of integer solutions to the famous equation we know as Pythagoras’ theorem, namely x2 + y2 = z2 (putting 3, 4, 5, say, for x, y, z); the Greeks proved that the prime numbers were as unlimited as the numbers themselves. The subject revived in the 16th century, with the discovery and translation from Greek of Diophantus’ Arithmetica. One of the simpler problems Diophantus records asks for a method that would generate the triples of integers satisfying Pythagoras’ equation. Pierre de Fermat, a professional jurist, passionate part-time mathematician and author of many splendid number theorems, thought about the obvious extension of Diophantus’ question: could a cube, for example, 27 (= 33) or 1000 (= 103), be split into a sum of two cubes and, more generally, could any higher power be split into a sum of two whole numbers each of that power? In other words (or rather symbols), Fermat was interested in whether the equation xn + yn = zn could have any integer solutions when the exponent n was 3 or more. He decided that it couldn’t, that no whole numbers could be found to replace x, y and z, which would make the equation true for any value of n other than 2. Notoriously, he didn’t give a proof. Instead, he apparently scribbled in the margin of his copy of Arithmetica that he had a proof but that it was too long to be written in the margin. That, at least, is the story: Fermat’s annotated copy of Diophantus has never been found.

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Letters

Vol. 19 No. 24 · 11 December 1997

Mathematics is not the organised hypocrisy glimpsed in the background of Brian Rotman’s self-portrait, ‘a renegade blowing the whistle’ (LRB, 27 November). Broadly speaking, if you accept the existence of the whole numbers 1, 2, 3 … as a set, some rules for making new sets from old ones and the use of the syllogism in arguing about them, then you have accepted modern mathematics as ‘true’. The simple is indivisible from the complex. If Fermat’s Last Theorem is metaphysical, then so is Baby’s First.

The serious worries are lower down. Style and standards of proof constantly drift: we visualise wrongly, we dupe ourselves, we miss possibilities. What keeps the literature from pullulating with errors is that falsehood tends to lead to more falsehood and finally to the absurd, such as that 1=2, from which we retrace our steps. But incomplete proofs of true state· ments are a legitimate concern. Wiles’s first draft was not quite there; in the late Eighties, an initially promising assault on the Poincare conjecture was lost altogether.

Why bother then? Why believe there are truths about numbers? What Wiles was doing in the attic all those years was raising what Wordsworth calls

that interminable building reared
By observation of infinities
In objects where no brotherhood exists
to passive minds.

Graham Nelson
Oxford

Vol. 20 No. 2 · 22 January 1998

One neglected reason for Andrew Wiles spending all those years wrestling Fermat’s Last Theorem to the attic floor (LRB, 27 November 1997) might be the excitement brought on by the resonance of the phrase itself: ‘Fermat’s Last Theorem’. A crucial part of such carefully shaped phrases is the use of the word ‘last’: there is, for instance, Trent’s Last Case (you can feel the tingly resonance there) or The Last of the Mohicans; in radio, Krapp’s Last Tape; in cinema, The Last Picture Show. (There are shaped phrases which carry a peculiar resonance and that don’t have ‘last’ in them, but there aren’t many – ‘trained by Jesuits’ and ‘the storming of the Winter Palace’.)

During World War Two a joke shop in Preston which had published Billy’s Weekly Liar for many years opportunistically trans – muted the organ into the Berlin Liar. The same shop sold matchbox coffins containing corps-es of miniature Hitlers. Opening the matchbox/coffin allowed Hitler’s penis to spring erect (the penises being made from snippets of the fine pink rubber tubing manufactured for bicycle tyre valves). These artefacts were labelled ‘Hitler’s Last Stand’. That the puzzle of Fermat’s Last Theorem was ‘solved’ by Wiles using 20th-century means has brought a sense of loss; hence the assertions that Fermat’s own solution (if it existed) must have been shorter and more elegant. The quest to re-create that solution (if it existed), and thus enable the thrill of the phrase to carry on through history, would require an imaginative leap into the past which all the mathematicians in all the gin joints in all the world couldn’t manage.

Leo Baxendale
Stroud, Gloucestershire

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