MATHEMATICS 



of them more than a hundred years old and yet 

 in which fundamental advances are being made. 

 One of the outposts is what is known as Fermat's 

 last theorem, which asserts that the equation 



x n + y = z n 



has no solutions whatever in which x, y, z, and n 

 are whole numbers, provided n is greater than 2. 

 It appears simple enough to understand the nature 

 of the theorem when n = 2 we have the right- 

 angled triangle whose sides can be x = 3, y == 4, 

 z = 5, or x = 12, y = 5, z = 13, and so on 

 but, though all tests made upon it when n is greater 

 than 2 indicate that it is correct, no strict proof 

 of it can be found. Fermat left many such 

 theorems, due probably to conjecture or repeated 

 trial, but this is now the sole survivor which resists 

 all attacks. A prize of 100,000 marks for a proof 

 or disproof of it was once founded at Vienna, and 

 has accumulated since. Hundreds of attempted 

 solutions are sent in, but all contain a fallacy. May 

 I call the attention of everybody to this theorem, 

 which is one towards the solution of which an 

 extensive mathematical training is of little help 

 a curious characteristic of all Fermat's theorems 

 and of many other of the more difficult problems 

 in the theory of numbers. 



There are many other such theorems for 

 example, one due to Euler, and more than 150 



'9 



