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An Essay on the Prize in Mathematical Sciences 2005

Pierre de Fermat was an eccentric jurist who lived in France during the 17th century, and whose real love was mathematics. His special love was the mysterious and beautiful properties of integers – the ordinary whole numbers of everyday arithmetic. Fermat published no scholarly papers, but he kept extensive notebooks in which one may find the beginnings of much of modern arithmetic.


While studying the ancient Greek text “Arithmetica” by Diofantus, Fermat came across the famous Pythagorean equation


a2 + b2 = c2


which students encounter in geometry in secondary school. Of special interest are right triangles whose side lengths are integers, such as


right triangles


It was known that there is in fact an infinite number of such triangles, and they may all be determined by elementary geometry. Fermat went on to study the equation


a3 + b3 = c3


and, more generally, for any integer n,


an + bn = cn.


He wrote in the margin of one of his notebooks that he had discovered his own “most wonderful proof” of the assertion that this equation has no integral solution for n = 3, 4, 5, …, and he provided only the news that the margin was too small to write it down. In fact, Fermat frequently made statements without giving proofs, to challenge or perhaps taunt his fellow mathematicians.


By the end of the 18th century all but the one written above of Fermat's theorems had been verified, and this then came to be known as Fermat's Last Theorem, or FLT. Although the FLT equation may look rather special, it has drawn the attention of many of the greatest mathematicians - in part because in studying it, one is led to unexpected and deep properties of ordinary whole numbers. By the late 20th century, FLT was known to be true for exponents of value to 4,000,000, but arithmetic is so unpredictable that the validity of FLT remained in doubt. In fact, a few mathematicians had suggested that FLT might be one of Gödels famous “undecidable propositions”, whose validity could not be established one way or the other.


After many years of little change, tectonic shifts in FLT began to occur in the mid-1980s through the work of Frey, Serre, Ribet and others, leading to the reformulation of the assertion as a mainstream conjecture of modern arithmetic. In other words, it had matured from a single puzzling equation into a significant and very active area of contemporary mathematics.


The reformulation concerned the elliptic curve


y2 = x(x - a)(x - b),


whose geometry leads deeply into a large part of the 19th and 20th century mathematics. If a, b, c are solutions to Fermat's equations, the reformulation was, in effect, that this particular elliptic curve should have very special symmetry properties. In mathematical terms it should be “modular”, which means that through a highly transcendental process it should be defined by a group of two by two matrices whose entries are roughly integral.


Enter Andrew John Wiles, who had been drawn to Fermat as a schoolboy, when he vowed that he would some day solve this famous problem. He had put aside his attempt when advised to pursue more mainstream research, work which placed him in the front rank of his contemporaries, but now he returned in earnest to FLT. After seven years of intense effort, in 1992 during a series of lectures at Cambridge University he announced a solution to FLT. A gap was found in the argument, which Wiles, together with Richard Taylor, was able to fill during the following year. He called his solution “the most important moment of my working life. It was so indescribably beautiful, it was so simple and elegant, and I just stared in disbelief for twenty minutes.”


Wiles' solution of FLT is perhaps the most remarkable single achievement in the field in recent times. In the depth and ingenuity of the arguments, and in the breadth of a vast array of modern mathematics – much of it from the last half century – that come into play, the solution to FLT stands as a landmark of contemporary mathematics. Not only did it settle an old problem that had not been resolved by many of the best mathematical minds for over 350 years; even more satisfying were the almost unimaginable beauty and intricacy of the arguments, which involved complex areas of number theory. This crowning achievement has opened vast new vistas for arithmeticians who continue to probe the mysteries of these most basic of mathematical objects, ordinary numbers.

Mathematical Sciences Selection Committee
The Shaw Prize

2 September 2005, Hong Kong