I feel a great honour in giving the Presentation Lecture for 2005 Laureate Professor Andrew WILES in Mathematical Sciences owing to his outstanding achievements in Number Theory and other domains of mathematics, particularly the monumental complete settlement of 265year-old FERMAT LAST THEOREM.

The FERMAT LAST THEOREM or FLT for short, had a long fascinating history. It asks whether there are positive integer solutions in (x,y,z) of the equation

xn + yn = zn(In)

for integer exponent n > 2. The case n = 2 is well-known already in remote antiquity. In fact, let x, y, z be the lengths of the two sides and the hypothenus of a right-angled triangle or a Gou-Gu Form in ancient China's terminology. Then equation (I2) is the famous Pythagorean equation or the Gou-Gu Theorem in China's antiquity. It is also well-known that the equation (I2) has positive integer solutions (x,y,z) = (3,4,5), (5,12,13), etc., see the accompanying figure below:

In fact, in the Chinese ancient classic¡m Nine Chapters of Arithmetic ¡nmore than 2000 years before it is already described that the whole set of possible positive integer solutions of (I2), with Gou, Gu, Xuan as the lengths of two sides and the hypothenus of a right-angled triangle or a Gou-Gu Form, is given by

x : y : z = Gou : Gu : Xuan = m 2 –
m 2 + n 2
2
 : m * n :
m 2 + n 2
2
(II)



Besides (x,y,z) = (3,4,5), (5,12,13) the¡m Nine Chapters ¡nlisted also the solutions (7,24,25), (8,15,17), (20,21,29), (48,55,73), (60,91,109), (20,99,101) corresponding to the case m : n = 2:1, 3:2, 4:3, 5:3,7:3, 11:5, 13:7, 11:9 respectively in the Formula (II ). Moreover, in the ¡m Annotations to Nine Chapters ¡n of scholar LIU Hui of 3c A.D. LIU had given a rigorous proof of (II ).

While the case n = 2 is elementary and understandable by school boys, the case n > 2 is entirely different. In year 1637 the French jurist Piere de Fermat, as an amateur in mathematics, announced that for n > 2 the equation (In) had no positive integer solutions at all. Fermat himself had proved the case n = 4 and in 17th century the great mathematician L. Euler had proved FLT for n = 3,5. However, the "lost" or actually "non-existant" proof of Fermat had puzzled mathematicians since that time. To "rediscover" or to "discover" a proof of FLT had attracted the greatest mathematical minds of the later years and centuries but resulted always in failure. However, in spite of these failures their efforts had brought out the fruits of giving impetus to some important mathematics disciplines and even naissance to new ones. Moreover, some new ideas introduced, the notion of ideals for example, had henceforth penetrated into the whole domain of mathematics and became henceforth indispensable tools for mathematical researches.

By the late 20th century it had been shown that FLT was known to be true for an infinity of exponents n and is true for 3 < n < 4,000,000. However, it is yet infinitely far from the ultimate goal: FLT should be true for all n > 3.

It comes now the final astonishing result of A.Wiles. Around ninetieth of last century, A.Wiles had already become reknown to mathematics circle owing to his dramatic achievements on many delicate problems in number theory. In the long run A.Wiles announced in year 1993 at the end of a series of lectures in Cambridge that the FLT had been thus proved. This not only completely solves the 265year-old problem but the method involved will influence the development of mathematics as a whole. In particular it had a deep impact to the important subject of algebraic geometry.

The announcement causes a thunder-like shock to the whole of mathematical society and a wide range of amateurs. The whole paper appeared in Annals of Mathematics, vol.141, (1995), of length about 110 pages.

It is for such Olympian accomplishments that we have the honour of declaring that the 2005 Shaw Prize in mathematical sciences is offered to

Professor Andrew WILES !!!