Pierre de Fermat (1601 - 1665)

How did a provincial lawyer, who provided few proofs and published only one anonymous piece become regarded as one of the greatest mathematicians?

And why did this amateur mathematician set so many people - amateur and professional mathematicians alike - on a 300 year hunt to prove a theorem?

Fermat’s life was bounded by obscurity. Mathematics was a hobby for him, to be fitted in after his working hours.

He lived his entire life in South-Western France and didn’t meet the great minds of his time. His only knowledge of the world beyond was a correspondence with people like Pascal and Huygens and a brief argument with Descartes whom he claimed was ‘groping about in the shadows’ over his law of refraction.

It is hard to trace his work, we have only his correspondence and his son’s posthumous publication of his work. He also insisted on working using Viete’s number system which Descartes had rendered obsolete.

This did not matter much in his own field of number theory, but meant that he did not have as many correspondents as he otherwise would have.

His mathematical discoveries were extraordinary. It seems that independently of Descartes he made the link of applying algebra to geometry through a co-ordinate system leading to what we now know of as Cartesian geometry.

He worked on differential calculus and is jointly credited with Pascal for founding the study of probability.

But his greatest efforts were reserved for number theory. Number Theory has always been regarded as difficult - in fact its name is a misnomer, since there is little theory, more a collection of haphazard results and one-off techniques.

Sometimes one thing works and then another; rarely is there a tool that can be systematically applied. This was certainly true in Fermat’s time, less so after the giants Euler and Gauss had left their profound marks in this area.

Fermat’s mathematics

Fermat stumbled on a technique that does work consistently - but only for a rather limited class of problems. It sounds exotic and profound - it is called the method of infinite descent. We will apply it to a particular problem, Suppose we are asked to find all the solutions in positive integers of the equation,

x^3+3y^3=9z^3.

(This is deliberately chosen to resemble his more famous ‘Theorem’. Such problems - problems demanding whole numbers as answers - are called Diophantine problems after the Greek mathematician who first wrote about them.)

We start off by assuming that a solution does exist. So that we can find positive, whole numbers a, b and c so that:

a^3+3b^3=9c^3.

Now the infinite descent begins! 3 divides the right hand side, and also one of the terms on the left. So it must also divide a^3 and in turn it must divide a itself - just think about it. Let us write this in the form

a=3a_1

and note that a_1 is smaller than a. Now we put this back in the equation, and find,

27a_1^3+3b^3=9c^3.

We can cancel a factor of 3 and are then left with,

9a_1^3+b^3=3c^3

Now we repeat the process for b. It must be divisible by 3 and so it can written in the form,

b=3b_1

in which b_1 is smaller than b. We put this in the equation:

9a_1^3+27b_1^3=3c^3

a 3 again cancels and we are left with:

3a_1^3+9b_1^3=c^3.

You’ve guessed it - c is also divisible by 3 and so can be written:

c=3c_1

in which c_1 is smaller than c. Put this into the equation one more time,

3a_1^3+9b_1^3=27c_1^3

and would you believe it, but after cancellation we are left with,

a_1^3+3b_1^3=9c_1^3

and that’s exactly the equation we started with. But this time the solutions a_1,b_1 and c_1 are each smaller than the first solution we started with. But as we are back at our starting point we could repeat the whole process - again and again, finding solutions that are smaller and smaller. But that is impossible for positive whole numbers - you can’t descend infinitely from them and keep on getting positive whole numbers. Ultimately they become negative.

This impossibility forces us to reject our starting assumption that there was such a solution and our proof is complete.

We can ‘see’ what this equation looks like by using Cartesian geometry. This is part of the 3-dimensional graph of the equation x^3+3y^3=9z^3:

Most famously, Fermat is known for his last theorem. He never published his Theorem, nor communicated it to others - it was discovered after his death. The discovery was made by his son in one of his books. In the margin of this book he had stated his theorem and gone on to say, ‘I have discovered a truly remarkable proof which this margin is too small to contain’. Mathematicians have been pulling the same trick ever since - at least students in examinations have. When stuck, just write, “I have the solution to this problem, but the margin of your answer sheet is too small to contain it.”.

The theorem concerns a Diophantine problem: that the equation

x^n+y^n=z^n

has no integer solutions for n > 2.

It is a generalisation of the Pythagorean equation

x^2+y^2=z^2

which has an infinity of solutions - 3,4 and 5; 5,12 and 13 for example. But the generalisation is that it no longer works and that is the truly remarkable fact and character of the result. It looks surprisingly simple - do not be deceived, in the recent proof by Professor Sir Andrew Wiles, it takes over 1,000 pages of very complicated mathematics. Fermat’s own method of infinite descent certainly doesn’t work, and neither does any other elementary approach. Despite this the proofs still roll in. Professor Landau - a German mathematician in Göttingen at the beginning of the last century - had postcards printed which he got his research students to send off to hopefuls. The postcard was printed ” Dear ……. the first mistake in your proof of Fermat’s Last Theorem occurs on line ….. of page ….” !

It is now believed that Fermat did not have a correct proof. But what a glorious mistake it was and what a glorious 300 year scramble it became. Along the way, by default, mathematics won out.