49 Fermat’s last theorem
We can add two square numbers together to make a third square. For instance, 5^2 +
122 = 13^2. But can we add two cubed numbers together to make another cube? What
about higher powers? Remarkably, we cannot. Fermat’s last theorem says that for any
four whole numbers, x, y, z and n, there are no solutions to the equation xn + yn = zn
when n is bigger than 2. Fermat claimed he’d found a ‘wonderful proof’, tantalizing the
generations of mathematicians that followed including a ten-year-old boy who read
about this mathematical treasure hunt one day in his local library.
Fermat’s last theorem is about a Diophantine equation, the kind of equation
which poses the stiffest of all challenges. These equations demand that their
solutions be whole numbers. They are named after Diophantus of Alexandria
whose Arithmetica became a milestone in the theory of numbers. Pierre de
Fermat was a 17th-century lawyer and government official in Toulouse in France.
A versatile mathematician, he enjoyed a high reputation in the theory of
numbers, and is most notably remembered for the statement of the last theorem,
his final contribution to mathematics. Fermat proved it, or thought he had, and
he wrote in his copy of Diophantus’s Arithmetica ‘I have discovered a truly
wonderful proof, but the margin is too small to contain it.’
Fermat solved many outstanding problems, but it seems that Fermat’s last
theorem was not one of them. The theorem has occupied legions of
mathematicians for three hundred years, and has only recently been proved. This
proof could not be written in any margin and the modern techniques required to
generate it throw extreme doubt on Fermat’s claim.
The equation x + y = z
How can we solve this equation in three variables x, y and z? In an equation
we usually have one unknown x but here we have three. Actually this makes the
equation x + y = z quite easy to solve. We can choose the values of x and y any
way we wish, add them to get z and these three values will give a solution. It is
as simple as that.
For example if we choose x = 3 and y = 7, the values x = 3, y = 7 and z = 10