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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