Figuring out Fermat

 

HIS is probably the best popular account of a scientific topic that I have ever read. It recounts the history of a succession of people dedicated to the pursuit of mathematical knowledge for its own sake - dedication which is illustrated by the legend of the death of Archimedes. A soldier of the invading Roman army approached and roughly questioned Archimedes. Archimedes was so preoccupied in studying a geometrical figure drawn in the sand that he didn't answer, whereupon the soldier speared him to death.

The book tells the tale of the quest to solve Fermat's Last Theorem, the most famous riddle in mathematics. In order to develop his narrative the author, in effect, runs through the history of mathematics, in particular the development of number theory. The story is told with a light and graceful touch and is peppered with fascinating characters, so that the reader is irresistibly tempted to turn the page to follow the next twist.

Mathematical ideas are of course, presented in abundance; but in a simple and easily understandable manner. An appendix presents some worked out examples of theorems for those who wish to go into things a little deeper.

Any schoolchild can understand Fermat's Last Theorem, but its solution eluded the best mathematical minds for 358 years. The theorem was proposed by the French amateur mathematician Pierre de Fermat in 1637. It is related to the most famous theorem in geometry - Pythagoras Theorem. This states that, in a right angled triangle, the square on the side opposite the right angle (hypotenuse) is equal to the sum of the squares on the other two sides. If we denote the length of the hypotenuse as Z, and the lengths of the other two sides as X and Y respectively, the theorem can be written: X 2 + Y 2= Z 2.

An infinitely large number of values of X, Y and Z satisfy this equation, e.g. 3 2+ 4 2 = 5 2 Fermat's Last Theorem says that, in the general equation Xn + Yn = Zn, where N is a whole number, no solutions exist except when n=2. Fermat declared that he had worked out a wonderful proof of his theorem, but he never published this proof.

The book traces the history of mathematics and number theory, from Pythagoras through Archimedes, Euclid, Diophantus, the Arabian and Indian period, Leonhard Euler, Sophie Germain, Ernst Kunimer, Carl Gauss, David Hilbert, Bertrand Russell, Kurt Gudel, Alan Turing, Yutaka Taniyama and Goro Shimura to the British mathematician Andrew Wiles, who finally solved Fermat's Last Theorem in 1994.

The book is full of stories about famous mathematicians like Leonhard Euler (1707-1783), probably the most prolific mathematician ever. He once worked at the court of Catherine the Great, where some French philosophers of an atheistic persuasion were wont to amuse themselves by arguing with courtiers, fatally undermining their faith in God. Catherine asked Euler to defend God in a debate with these philosophers. Euler asked for a blackboard and wrote - "(X+Y)2 = X2 + 2XY + Y2 - therefore God exists. Reply." Euler realised that the philosophers knew little algebra or geometry, and, more importantly, that they were too vain to admit this. The philosophers looked at the board, nodded their heads and walked away.

Andrew Wiles was fascinated by Fermat's Last Theorem from the age of 10 and in 1986 began a full time effort to solve it. He literally locked himself away in an attic study and slaved away; after eight years of intensive effort he announced his solution at a lecture in Cambridge, to wild enthusiasm from the mathematical world and significant media attention.

SOLVING such mathematical problems is not merely some grandiose version of solving a crossword puzzle; their value lies in the interesting mathematical ideas that the work throws up. For example, in proving Fermat's Last Theorem, Wiles also proved another mathematical conjecture - the Taniyama Shimura conjecture - which unites two branches of mathematics previously thought to be entirely separate.

Fermat's Last Theorem also illustrates how incredibly hard the dedicated researcher works. For example, Euler once worked so hard on a problem over a week that he lost the sight in one eye. I heartily recommend this book to all - and particularly to those who think that academic researchers spend their entire summers scratching their backsides on holidays, and most of the rest of the year scratching them at work.