Fermat’s Last Theorem. Simon Singh

building block too many or too few. For example, if we begin with the cubes 6³ (x³) and 8³ (y³) and rearrange the building blocks, then we are only one short of making a complete 9 × 9 × 9 cube, as shown in Figure 4.

      Finding three numbers which fit the cubed equation perfectly seems to be impossible. That is to say, there appear to be no whole number solutions to the equation

      

      Furthermore, if the power is changed from 3 (cubed) to any higher number n (i.e. 4, 5, 6, …), then finding a solution still seems to be impossible. There appear to be no whole number solutions to the more general equation

      By merely changing the 2 in Pythagoras’ equation to any higher number, finding whole number solutions turns from being relatively simple to being mind-bogglingly difficult. In fact, the great seventeenth-century Frenchman Pierre de Fermat made the astonishing claim that the reason why nobody could find any solutions was that no solutions existed.

      Fermat was one of the most brilliant and intriguing mathematicians in history. He could not have checked the infinity of numbers, but he was absolutely sure that no combination existed which would fit the equation perfectly because his claim was based on proof. Like Pythagoras, who did not have to check every triangle to demonstrate the validity of his theorem, Fermat did not have to check every number to show the validity of his theorem. Fermat’s Last Theorem, as it is known, stated that

      

      has no whole number solutions for n greater than 2.

      As Wiles read each chapter of Bell’s book, he learnt how Fermat had become fascinated by Pythagoras’ work and had eventually come to study the perverted form of Pythagoras’ equation. He then read how Fermat had claimed that even if all the mathematicians in the world spent eternity looking for a solution to the equation they would fail to find one. He must have eagerly turned the pages, relishing the thought of examining the proof of Fermat’s Last Theorem. However, the proof was not there. It was not anywhere. Bell ended the book by stating that the proof had been lost long ago. There was no hint of what it might have been, no clues as to the proof’s construction or derivation. Wiles found himself puzzled, infuriated and intrigued. He was in good company.

      For over 300 years many of the greatest mathematicians had tried to rediscover Fermat’s lost proof and failed. As each generation failed, the next became even more frustrated and determined. In 1742, almost a century after Fermat’s death, the Swiss mathematician Leonhard Euler asked his friend Clêrot to search Fermat’s house in case some vital scrap of paper still remained. No clues were ever found as to what Fermat’s proof might have been. In Chapter 2 we shall find out more about the mysterious Pierre de Fermat and how his theorem came to be lost, but for the time being it is enough to know that Fermat’s Last Theorem, a problem that had captivated mathematicians for centuries, had captured the imagination of the young Andrew Wiles.

      Sat in Milton Road Library was a ten-year-old boy staring at the most infamous problem in mathematics. Usually half the difficulty in a mathematics problem is understanding the question, but in this case it was simple – prove that xⁿ + yⁿ = zⁿ has no whole number solutions for n greater than 2. Andrew was not daunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. He immediately set to work using all his textbook techniques to try and recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world.

      Thirty years later Andrew Wiles was ready. Standing in the auditorium of the Isaac Newton Institute, he scribbled on the board and then, struggling to contain his glee, stared at his audience. The lecture was reaching its climax and the audience knew it. One or two of them had smuggled cameras into the lecture room and flashes peppered his concluding remarks.

      With the chalk in his hand he turned to the board for the last time. The final few lines of logic completed the proof. For the first time in over three centuries Fermat’s challenge had been met. A few more cameras flashed to capture the historic moment. Wiles wrote up the statement of Fermat’s Last Theorem, turned towards the audience, and said modestly: ‘I think I’ll stop here.’

      Two hundred mathematicians clapped and cheered in celebration. Even those who had anticipated the result grinned in disbelief. After three decades Andrew Wiles believed he had achieved his dream, and after seven years of isolation he could reveal his secret calculation. However, while euphoria filled the Newton Institute tragedy was about to strike. As Wiles was enjoying the moment, he, along with everyone else in the room, was oblivious of the horrors to come.

       2 The Riddler

      ‘Do you know,’ the Devil confided, ‘not even the best mathematicians on other planets – all far ahead of yours – have solved it? Why, there’s a chap on Saturn – he looks something like a mushroom on stilts – who solves partial differential equations mentally; and even he’s given up.’

      Arthur Porges, ‘The Devil and Simon Flagg’

      

      Pierre de Fermat was born on 20 August 1601 in the town of Beaumont-de-Lomagne in south-west France. Fermat’s father, Dominique Fermat, was a wealthy leather merchant, and so Pierre was fortunate enough to enjoy a privileged education at the Franciscan monastery of Grandselve, followed by a stint at the University of Toulouse. There is no record of the young Fermat showing any particular brilliance in mathematics.

      Pressure from his family steered Fermat towards a career in the civil service, and in 1631 he was appointed conseiller au Parlement de Toulouse, a councillor at the Chamber of Petitions. If locals wanted to petition the King on any matter they first had to convince Fermat or one of his associates of the importance of their request. The councillors provided the vital link between the province and Paris. As well as liaising between the locals and the monarch, the councillors made sure that royal decrees emanating from the capital were implemented back in the regions. Fermat was an efficient civil servant, who by all accounts carried out his duties in a considerate and merciful manner.

      Fermat’s additional duties included service in the judiciary and he was senior enough to deal with the most severe cases. An account of his work is given by the English mathematician, Sir Kenelm Digby. Digby had requested to see Fermat, but in a letter to a mutual colleague, John Wallis, he reveals that the Frenchman had been occupied with pressing judicial matters, thus excluding the possibility of a meeting:

      

      It is true that I had exactly hit the date of the displacement of the judges of Castres to Toulouse, where he [Fermat] is the Supreme Judge to the Sovereign Court of Parliament; and since then he has been occupied with capital cases of great importance, in which he has finished by imposing a sentence that has made a great stir; it concerned the condemnation of a priest, who had abused his functions, to be burned at the stake. This affair has just finished and the execution has followed.

      

      Fermat corresponded regularly with Digby and Wallis. Later we will see that the letters were often less than friendly, but they provide vital insights into Fermat’s daily life, including his academic work.

      Fermat rose rapidly within the ranks of the civil service and became a member of the social élite, entitling him to use de as part of his name. His promotion was not necessarily the result of ambition, but rather a matter of health. The plague