Most difficult puzzle solved

A BRITISH mathematician working at Princeton University in the United States has solved the most famous and most tantalising puzzle in the history of science.

Professor Andrew Wiles told an audience of mathematicians at Cambridge University yesterday that he had proved 'Fermat's last theorem'.

It is something scientists have been trying to do for more than three centuries. It took Professor Wiles three hours of successive lectures, starting on Monday, to explain how he had done so.

The solution involves the use of mathematical functions defined using 'imaginary' numbers - the square root of minus one. Appropriately also, Professor Wiles kept his audience waiting until the very end before announcing that the solution followed almost as a by-product of the more difficult theorems which he had just proved to them.

Fermat's puzzle builds on Pythagoras' theorem which says that a triangle whose sides are 3, 4 and 5 units long must be right- angled. This follows from Pythagoras because the sum of 3 squared and 4 squared (9 + 16) is 25 - which is 5 squared.

Pierre de Fermat, a French mathematician who lived from 1601 to 1665, decided that it was not possible to find similar whole-number solutions if he considered cubes, instead of just the squares of numbers. In fact, he decided that although there are literally infinitely many whole-number solutions of the equations

x + y = z, and

x2 + y2 = z2 ,

there are no whole-number solutions at all of the equation

xn + yn = zn

where n is greater than 2.

Fermat wrote in the margin of his notebook that he had 'found an admirable proof of this theorem, but the margin is too narrow to contain it'.

Because the theorem is comparatively simple to state, many amateurs as well as professional mathematicians have tried to solve it over the centuries. The amateurs always got it wrong. While the professionals did not do much better until now, their efforts opened up many deep and productive areas of mathematical theory.