Fermat's Last Theorem (26 page)

Figure 14. In 5-clock arithmetic the number line is truncated at 5 and looped back on itself. The number 5 coincides with 0, and therefore is replaced by it.

The solutions are:

Although some of these solutions would not be valid in normal arithmetic, in 5-clock arithmetic they are acceptable. For example, the fourth solution (
x
= 1,
y
= 4) works as follows:

But remember, 20 is equivalent to 0 in 5-clock arithmetic, because 5 will divide into 20 with a remainder of 0.

Because they could not list all the solutions to an elliptic equation working in infinite space, mathematicians, including Wiles,
settled for working out the number of solutions in all the different clock arithmetics. For the elliptic equation given above the number of solutions in 5-clock arithmetic is four, and so mathematicians say
E
₅
= 4. The number of solutions in other clock arithmetics can also be calculated. For example, in 7-clock arithmetic the number of solutions is nine, and so
E
₇
= 9.

To summarise their results, mathematicians list the number of solutions in each clock arithmetic and call this list the
L
-series for the elliptic equation. What the
L
stands for has been long forgotten although some have suggested that it is the
L
of Gustav Lejeune-Dirichlet, who worked on elliptic equations. For clarity I will use the term
E
-series – the series that is derived from an elliptic equation. For the example given above the
E
-series is as follows:

Because mathematicians cannot say how many solutions some elliptic equations have in normal number space which extends up to infinity, the
E
-series appears to be next best thing. In fact the
E
-series encapsulates a great deal of information about the elliptic
equation it describes. In the same way that biological DNA carries all the information required to construct a living organism, the
E
-series carries the essence of the elliptic equation. The hope was that by studying the
E
-series, this mathematical DNA, mathematicians would ultimately be able to calculate everything they could ever wish to know about an elliptic equation.

Working alongside John Coates, Wiles rapidly established his reputation as a brilliant number theorist with a profound understanding of elliptic equations and their
E
-series. As each new result was achieved and each paper published, Wiles did not realise that he was gathering the experience which would many years later bring him to the verge of a proof for Fermat's Last Theorem.

Although nobody was aware of it at the time, the mathematicians of post-war Japan had already triggered a chain of events which would inextricably link elliptic equations to Fermat's Last Theorem. By encouraging Wiles to study elliptic equations, Coates had given him the tools which would later enable him to work on his dream.

The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

G.H. Hardy

In the January of 1954 a talented young mathematician at the University of Tokyo paid a routine visit to his departmental library. Goro Shimura was in search of a copy of
Mathematische Annalen
, Vol. 24. In particular he was after a paper by Deuring on his algebraic theory of complex multiplication, which he needed in order to help him with a particularly awkward and esoteric calculation.

To his surprise and dismay, the volume was already out. The borrower was Yutaka Taniyama, a vague acquaintance of Shimura who lived on the other side of the campus. Shimura wrote to Taniyama explaining that he urgently needed the journal to complete the nasty calculation, and politely asked when it would be returned.

A few days later, a postcard landed on Shimura's desk. Taniyama had replied, saying that he too was working on the exact same calculation and was stuck at the same point in the logic. He suggested that they share their ideas and perhaps collaborate on the problem. This chance encounter over a library book ignited a partnership which would change the course of mathematical history.

Taniyama was born on 12 November 1927 in a small town a few miles north of Tokyo. The Japanese character symbolising his first name was intended to read ‘Toyo', but most people outside his family misinterpreted it as ‘Yutaka', and as Taniyama grew up he accepted and adopted this title. As a child Taniyama's education was constantly interrupted. He suffered several bouts of ill health, and during his teenage years he was struck down by tuberculosis and had to miss two years of high school. The onset of war caused even greater disruption to his schooling.

Goro Shimura, one year younger than Taniyama, had his education stopped altogether during the war years. His school was shut down and, instead of attending lessons, Shimura had to help the war effort by working in a factory assembling aircraft parts. Each evening he would attempt to make up for his lost schooling and in particular found himself drawn to mathematics. ‘Of course there are many subjects to learn, but mathematics was the easiest because I could simply read mathematical textbooks. I learnt calculus by reading books. If I'd wanted to pursue chemistry or physics then I would have needed scientific equipment and I had no access to such things. I never thought that I was talented. I was just curious.'

A few years after the war had finished, Shimura and Taniyama found themselves at university. By the time they had exchanged postcards over the library book, life in Tokyo was beginning to return to normal and the two young academics could afford one or two small luxuries. They spent their afternoons in the coffee-shops, in the evenings they dined in a little restaurant specialising in whale
meat, and at weekends they would stroll through the botanical gardens or the city park. All ideal locations for discussing their latest mathematical thoughts.

Although Shimura had a whimsical streak – even today he retains his fondness for Zen jokes – he was far more conservative and conventional than his intellectual partner. Shimura would rise at dawn and immediately get down to work, whereas his colleague would often still be awake at this time, having worked through the night. Visitors to his apartment would often find Taniyama fast asleep in the middle of the afternoon.

While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surprisingly this was a trait that Shimura admired: ‘He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes.'

Taniyama was the epitome of the absent-minded genius and this was reflected in his appearance. He was incapable of tying a decent knot, and so he decided that rather than tie his shoelaces a dozen times a day he would not tie them at all. He would always wear the same peculiar green suit with a strange metallic sheen. It was made from a fabric which was so outrageous that it had been rejected by the other members of his family.

When they met in 1954 Taniyama and Shimura were just beginning their mathematical careers. The tradition was, and still is, for young researchers to be taken under the wing of a professor who would guide the fledgling brain, but Taniyama and Shimura rejected this form of apprenticeship. During the war real research had ground to a halt and even by the 1950s the mathematics faculty had still not recovered. According to Shimura, the professors were ‘tired, jaded and disillusioned'. In comparison the post-war
students were passionate and eager to learn, and they soon realised that the only way forward would be for them to teach themselves. The students organised regular seminars, taking it in turn to inform each other of the latest techniques and breakthroughs. Despite his otherwise lackadaisical attitude, when it came to the seminars Taniyama provided a ferocious driving force. He would encourage the more senior students to explore uncharted territory, and for the younger students he acted as a father figure.

Because of their isolation, the seminars would occasionally cover subjects which were generally considered passe in Europe and America. The students' naïvety meant that they studied equations which had been abandoned in the West. One particularly unfashionable topic which fascinated both Taniyama and Shimura was the study of
modular forms.

Modular forms are some of the weirdest and most wonderful objects in mathematics. They are one of the most esoteric entities in mathematics and yet the twentieth-century number theorist Martin Eichler rated them as one of the five fundamental operations: addition, subtraction, multiplication, division and modular forms. Most mathematicians would consider themselves masters of the first four operations, but the fifth one they still find a little confusing.

The key feature of modular forms is their inordinate level of symmetry. Although most people are familiar with the everyday concept of symmetry, it has a very particular meaning in mathematics, which is that an object has symmetry if it can be transformed in a particular way and yet afterwards appear to be unchanged. To appreciate the immense symmetry of a modular form it helps to first examine the symmetry of a more mundane object such as a simple square.

Figure 15. A simple square exhibits both rotational and reflectional symmetry.

In the case of a square, one form of symmetry is rotational. That is to say, if we imagine a pivot at the point where the
x
-axis and
y
-axis cross, then the square in
Figure 15
can be rotated by one quarter of a turn, and afterwards it will appear to be unchanged. Similarly, rotations by half a turn, three-quarters of a turn and one full turn will also leave the square apparently unchanged.

In addition to rotational symmetry the square also possesses reflectional symmetry. If we imagine a mirror placed along the
x
-axis then the top half of the square would reflect exactly onto the lower half, and vice versa, so after the transformation the square would appear to remain unchanged. Similarly we can define three other mirrors (along the
y
-axis and along the two diagonals) for which the reflected square would appear to be identical to the original one.