Fermat’s Last Theorem. Simon Singh

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СКАЧАТЬ inventions to patent, the mathematics department of any university is the least secretive of all. The community prides itself in an open and free exchange of ideas and tea-time breaks have evolved into daily rituals during which concepts are shared and explored over biscuits and Earl Grey. As a result it is increasingly common to find papers being published by co-authors or teams of mathematicians and consequently the glory is shared out equally. However, if Professor Wiles had genuinely discovered a complete and accurate proof of Fermat’s Last Theorem, then the most wanted prize in mathematics was his and his alone. The price he had to pay for his secrecy was that he had not previously discussed or tested any of his ideas with the mathematics community and therefore there was a significant chance that he had made some fundamental error.

      Ideally Wiles had wanted to spend more time going over his work to allow him to check fully his final manuscript. Then the unique opportunity arose to announce his discovery at the Isaac Newton Institute in Cambridge and he abandoned caution. The sole aim of the institute’s existence is to bring together the world’s greatest intellects for a few weeks in order to hold seminars on a cutting-edge research topic of their choice. Situated on the outskirts of the university, away from students and other distractions, the building is especially designed to encourage the academics to concentrate on collaboration and brainstorming. There are no dead-end corridors in which to hide and every office faces a central forum. The mathematicians are supposed to spend time in this open area, and are discouraged from keeping their office doors closed. Collaboration while moving around the institute is also encouraged – even the elevator, which only travels three floors, contains a blackboard. In fact every room in the building has at least one blackboard, including the bathrooms. On this occasion the seminars at the Newton Institute came under the heading of ‘L-functions and Arithmetic’. All the world’s top number theorists had been gathered together in order to discuss problems relating to this highly specialised area of pure mathematics, but only Wiles realised that L-functions might hold the key to solving Fermat’s Last Theorem.

      Although he had been attracted by having the opportunity to reveal his work to such an eminent audience, the main reason for making the announcement at the Newton Institute was that it was in his home town, Cambridge. This was where Wiles had been born, it was here he grew up and developed his passion for numbers, and it was in Cambridge that he had alighted on the problem which was to dominate the rest of his life.

      The Last Problem

      In 1963, when he was ten years old, Andrew Wiles was already fascinated by mathematics. ‘I loved doing the problems in school, I’d take them home and make up new ones of my own. But the best problem I ever found I discovered in my local library.’

      One day, while wandering home from school, young Wiles decided to visit the library in Milton Road. It was rather impoverished compared with the libraries of the colleges, but nonetheless it had a generous collection of puzzle books and this is what often caught Andrew’s attention. These books were packed with all sorts of scientific conundrums and mathematical riddles, and for each question the solution would be conveniently laid out somewhere in the final few pages. But this time Andrew was drawn to a book with only one problem, and no solution.

      The book was The Last Problem by Eric Temple Bell, the history of a mathematical problem which has its roots in ancient Greece, but which only reached full maturity in the seventeenth century. It was then that the great French mathematician Pierre de Fermat inadvertently set it as a challenge for the rest of the world. One great mathematician after another had been humbled by Fermat’s legacy and for three hundred years nobody had been able to solve it. There are other unsolved questions in mathematics, but what makes Fermat’s problem so special is its deceptive simplicity. Thirty years after first reading Bell’s account, Wiles told me how he felt the moment he was introduced to Fermat’s Last Theorem: ‘It looked so simple, and yet all the great mathematicians in history couldn’t solve it. Here was a problem that I, a ten-year-old, could understand and I knew from that moment that I would never let it go. I had to solve it.’

      The problem looks so straightforward because it is based on the one piece of mathematics that everyone can remember – Pythagoras’ theorem:

      

      In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

      

      As a result of this Pythagorean ditty, the theorem has been scorched into millions if not billions of human brains. It is the fundamental theorem that every innocent schoolchild is forced to learn. But despite the fact that it can be understood by a ten-year-old, Pythagoras’ creation was the inspiration for a problem which had thwarted the greatest mathematical minds of history.

      Pythagoras of Samos was one of the most influential and yet mysterious figures in mathematics. Because there are no first-hand accounts of his life and work, he is shrouded in myth and legend, making it difficult for historians to separate fact from fiction. What seems certain is that Pythagoras developed the idea of numerical logic and was responsible for the first golden age of mathematics. Thanks to his genius numbers were no longer merely used to count and calculate, but were appreciated in their own right. He studied the properties of particular numbers, the relationships between them and the patterns they formed. He realised that numbers exist independently of the tangible world and therefore their study was untainted by the inaccuracies of perception. This meant he could discover truths which were independent of opinion or prejudice and which were more absolute than any previous knowledge.

      Living in the sixth century BC, Pythagoras gained his mathematical skills on his travels throughout the ancient world. Some tales would have us believe that he travelled as far as India and Britain, but what is more certain is that he gathered many mathematical techniques and tools from the Egyptians and Babylonians. Both these ancient peoples had gone beyond the limits of simple counting and were capable of performing complex calculations which enabled them to create sophisticated accounting systems and construct elaborate buildings. Indeed they saw mathematics as merely a tool for solving practical problems; the motivation behind discovering some of the basic rules of geometry was to allow reconstruction of field boundaries which were lost in the annual flooding of the Nile. The word itself, geometry, means ‘to measure the earth’.

      Pythagoras observed that the Egyptians and Babylonians conducted each calculation in the form of a recipe which could be followed blindly. The recipes, which would have been passed down through the generations, always gave the correct answer and so nobody bothered to question them or explore the logic underlying the equations. What was important for these civilisations was that a calculation worked – why it worked was irrelevant.

      After twenty years of travel Pythagoras had assimilated all the mathematical rules in the known world. He set sail for his home island of Samos in the Aegean Sea with the intention of founding a school devoted to the study of philosophy and in particular concerned with research into his newly acquired mathematical rules. He wanted to understand numbers, not merely exploit them. He hoped to find a plentiful supply of free-thinking students who could help him develop radical new philosophies, but during his absence the tyrant Polycrates had turned the once liberal Samos into an intolerant and conservative society. Polycrates invited Pythagoras to join his court, but the philosopher realised that this was only a manoeuvre aimed at silencing him and therefore declined the honour. Instead he left the city in favour of a cave in a remote part of the island, where he could contemplate without fear of persecution.

      Pythagoras did not relish his isolation and eventually resorted to bribing a young boy to be his first pupil. The identity of the young boy is uncertain but some historians have suggested that his name was also Pythagoras, and that the student would later gain fame as the first person to suggest that athletes should eat meat to improve their physique. Pythagoras, the teacher, paid his student three oboli for each lesson he attended and noticed СКАЧАТЬ