Unravelling the rubric of the cube

Two books attempt a down-to-earth explanation of mathematical theory. Ian Stewart sums them up; Fermat's Last Theorem by Simon Singh, Fourth Estate, pounds 12.99 Fermat's Last Theorem by Amir Aczel, Viking, pounds 9.99
Somewhere around 1637, the lawyer Pierre de Fermat penned the most notorious marginal note in the history of mathematics. "It is impossible for a cube to be written as a sum of two cubes, or a fourth power as the sum of two fourth powers, or, in general, any number that is a power greater than the second to be written as a sum of two like powers." By the 1990s, Fermat's statement, now known as Fermat's Last Theorem, was still unproved. Generations of mathematicians had given it their best shot, and all had failed. Along the way they had shown that the theorem was true for all powers up to the 250,000th or so, but nobody had put the problem to rest.

On 23 June 1993, Andrew Wiles - a diffident English-born mathematician working at Princeton - gave the last of three lectures at the Isaac Newton Institute in Cambridge. Towards the end, Wiles outlined a partial proof of a central problem in number theory, the Taniyama-Shimura Conjecture. "And this proves Fermat's Last Theorem," he added. "I think I'll stop there".

Wiles's dramatic story has been told in a marvellous BBC television programme made for Horizon by John Lynch. It nearly didn't happen, for in mid-1993 the proof began to unravel. The usual plot-line consists of ever more frantic attempts to repair the flaw, degenerating into abject failure. Wiles avoided that plot. By a stroke of good fortune, allied to months of intense effort, he suddenly saw how to fix his proof so that it really worked.

Now we have two books that put the tale into print for the person in the street. One is by Simon Singh, who worked with Lynch; the other is by Amir Aczel, a Massachusetts statistician.

Both hook the reader's attention with Wiles's enigmatic announcement, and then put Fermat's problem into historical context. Neither offers more than allusions about what Wiles actually did - though on balance Singh gets closer. I don't blame them: there is absolutely no way to explain the nuts and bolts to non-specialists. Even experts find the ideas tough going. Instead, both authors sensibly opt to tell the story of the people whose ideas led up to Wiles's achievement, and to offer glimpses into his background and his motivation.

The tale must inevitably be spun as two virtually separate yarns. The first tails off ineffectually in the 19th century, after the epic ideas of Ernst Kummer had made it possible to tackle the theorem for a great many powers - but not all. The second picks up in the mid-1950s with the bold - and widely disregarded - conjecture of the Japanese mathematicians Goro Shimura and Yutaka Taniyama, the discovery by Gerhard Frey of its possible link to Fermat's Last Theorem, and the link's confirmation by Ken Ribet. Prove Taniyama-Shimura - "every elliptic curve is modular" - and Fermat inevitably follows. And that, pretty much, is what Wiles did.

Aczel is shorter, and unlike Singh he makes it clear that Wiles proved only part of the Taniyama-Shimura conjecture - the so called "semistable" case. But rather too much of Aczel's history comes straight from Eric Temple Bell's flawed classic Men of Mathematics. His pictures are too often of poor quality. A crucial illustration, captioned "Pierre de Fermat's Last Theorem as reproduced in an edition of Diophantus's Arithmerica published by Fermat's son Samuel", is nothing of the kind. It is a picture of the problem that motivated Fermat's conjecture - but his crucial comment, which in the edition edited by his son follows immediately afterwards, is omitted. Singh gets the right picture: comparing the page numbers, it looks as if Aczel's illustration has been taken from the wrong edition.

I especially disliked Aczel's attempt to inject controversy by suggesting that presenting research at a conference, before publishing it in a journal, is unconventional. As a mathematician, he must know that nowadays most research in the subject first sees the light of day at conferences and seminars. Publication usually takes years, and the mathematical community can't afford to wait that long.

Singh's history is far better researched, including lots of detail that is not readily available from standard sources. Singh also conveys a graphic impression of what the key personnel did, and why, based on interviews and discussions. His treatment of the relationship between Taniyama (who eventually killed himself, for no obvious reason) and Shimura is informative and moving. Unfortunately Singh spoils his ending with his own piece of phoney controversy - a spurious suggestion that Wiles's magnificent achievement somehow represents the final gasp of pencil-and-paper proofs.

I know exactly where he got that idea from: a now notorious article "The Death of Proof", written by John Horgan for Scientific American. It is notorious because virtually every mathematician Horgan interviewed has repudiated his conclusions. Horgan has now played the game for much higher stakes with The End of Science, a wonderfully readable book with a completely ridiculous theme.

Which Fermat should you buy? Both books are readable and enjoyable, and most people will not be bothered by their shortcomings. Neither is as definitive as I would have hoped. But Singh is better researched, better written, better illustrated - and only a third expensive at twice the length.