杏吧原创

IN all mathematics, perhaps only number theory appeals to the amateur and the professional alike. Dealing with such fundamental properties of numbers as primality or divisibility, number theory has a simplicity and beauty obvious to all but the innumerate. Yet this simplicity is deceptive. Take, for example, Euclid鈥檚 brilliantly elegant proof of the infinitude of primes, one of the earliest results in the history of number theory. Euclid鈥檚 approach was to set out with the assumption that there is only a finite number of primes, and then show that this leads to a contradiction.

It seems but a short step to go on to prove that there is an infinitude of prime pairs 鈥� that is, primes like 3 and 5, 17 and 19, 4967 and 4969 鈥� that are separated by a single composite number. Yet to this day no one has come up with a watertight proof of this 鈥渙bvious鈥� deduction.

Given the huge amount of effort already expended on this problem by professionals, the chances of some amateur coming up with a proof seem slim. But you never know: as the great Cambridge number theorist G. H. Hardy discovered in 1913, not all amateur mathematicians are yo-yos. In a scruffy package covered in stamps from India, he found a letter from a clerk in Madras and a collection of scribbled mathematical results. The author was Srinivasa Ramanujan, a mathematical genius who later joined Hardy in Cambridge to produce some of the most outstanding work in the history of number theory.

The most famous of all 鈥渁mateur鈥� number theorists is the French lawyer Pierre de Fermat (1601-1665). When not being interrupted by his parliamentary duties in Toulouse and Castres, Fermat pottered about in mathematics, producing results in areas as diverse as geometry and probability theory.

His most important work, however, is in number theory, where his results are used routinely by professionals to this day. His so-called 鈥淟ittle Theorem鈥�, for example, states that for any prime p and whole number n, subtracting n from n raised to the power p always leaves a number divisible by p, a result whose echoes can be heard today in fields as diverse as cryptography and group theory.

Until now, my own knowledge of Fermat鈥檚 life and career had come from E. T. Bell鈥檚 celebrated Men of Mathematics. From this notoriously romanticised biographical compendium, I learnt that there was not a lot to say about Fermat鈥檚 nonmathematical career, that his work in other areas of mathematics was pretty dull, and that he was in the habit of stating results, but not giving their proof. Most importantly I learn that Fermat was utterly reliable: in other words, he never claimed to have proved a theorem when he hadn鈥檛, but when he did, his theorems always turned out to be true.

Michael Mahoney鈥檚 book 鈥� a revised version of his 1973 original 鈥� gives us twenty-five times more material on Fermat than does Bell, but the basic story remains the same. Little can be said about Fermat鈥檚 personal and professional life because there are so few extant documents referring to it, and Fermat鈥檚 non-number theoretical work was indeed generally eclipsed by more gifted mathematicians.

So Mahoney is unable to tell us much more about Fermat the man, and the large amount of space Mahoney devotes to Fermat鈥檚 less impressive work in other areas will leave nonspecialists glassy-eyed.

Crucially, however, Mahoney disabuses us of Bell鈥檚 rose-tinted view of Fermat鈥檚 reliability. According to Bell, Fermat was convinced that numbers of the form ²ⁿ 2 raised to the power 2, raised to the power n, are always followed by a prime. Although this rule works for n = 0, 1, 2, 3 and 4, the Swiss mathematician and physicist Leonhard Euler showed in 1732 that it fails for n = 5. According to Bell, Fermat鈥檚 integrity or competence cannot be impeached for this, however, as Fermat only ever said that he was 鈥渃onvinced鈥� 鈥� not that he had a proof.

Not so: Mahoney reveals that in 1659 Fermat did claim that he had finally found a proof, obtained by his ingenious method of 鈥渋nfinite descent鈥� (a sort of reverse proof by induction).

The importance of this demonstration of Fermat鈥檚 occasional unreliability lies in its implications for the most famous result associated with his name: Fermat鈥檚 Last Theorem. In a notorious marginal note, Fermat announced that he had a 鈥渕arvellous demonstration鈥� that it was impossible to form the nth power of an integer by adding together the nth powers of two smaller integers for any value of n above 2. Mahoney argues that Fermat almost certainly based his supposed proof on his method of infinite descent. If true, it means Fermat was again either self-deceived or dishonest: the method fails for n greater than 22.

In June 1993, Andrew Wiles of Princeton University outlined a proof now widely considered to show Fermat was indeed right, albeit for the wrong reasons. The proof is of staggering complexity, exploiting techniques beyond anything available to Fermat. Frustratingly, Mahoney devotes precisely 32 words to this key breakthrough; indeed, he says next to nothing about any modern work that covers the Last Theorem. Admittedly, Wiles鈥檚 original proof had holes, but from the outset most mathematicians were sure these could be patched. Mahoney鈥檚 lack of courage here spoils what must otherwise be regarded as the definitive work on the patron saint of amateur mathematicians.