Fermat's Last Theorem (31 page)

In order to find a solution, Wiles adopted his usual approach to solving difficult problems. ‘I sometimes write scribbles or doodles. They're not important doodles, just subconscious doodles. I never use a computer.' In this case, as with many problems in number theory, computers would be of no use whatsoever. The Taniyama–Shimura conjecture applied to an infinite number of equations and, although a computer could check an individual case in a few seconds, it could never check all cases. Instead what was required was a logical step-by-step argument which would effectively give a reason and explain why every elliptic equation had to be modular. To find the proof Wiles relied solely on a piece of paper, a pencil and his mind. ‘I carried this thought around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.'

After a year of contemplation Wiles decided to adopt a general strategy known as
induction
as the basis for his proof. Induction is an immensely powerful form of proof, because it can allow a mathematician to prove that a statement is true for an infinite number of cases by only proving it for just one case. For example, imagine that a mathematician wants to prove that a statement is true for every counting number up to infinity. The first step is to prove that the statement is true for the number 1, which presumably is a fairly straightforward task. The next step is to show that if the statment is true for the number 1 then it must be true for the number 2, and if it is true for the number 2 then it must be true for the number 3, and if it is true for the number 3 then it must be true for the number 4, and so on. More generally, the mathematician has to show that if the statement is true for any number
n
, then it must be true for the next number
n
+ 1.

Proof by induction is essentially a two step process:

(1) Prove that the statement is true for the first case.

(2) Prove that if the statement is true for any one case, then it must be true for the next case.

Another way to think of proof by induction is to imagine the infinite number of cases as an infinite line of dominoes. In order to prove every case it is necessary to find a way of knocking down every one of the dominoes. Knocking them down one by one would take an infinite amount of time and effort, but proof by induction allows mathematicians to knock them all down by just knocking down the first one. If the dominoes are carefully arranged, then knocking down the first domino will knock down the second domino, which will in turn knock down the third domino, and so on to infinity. Proof by induction invokes the domino effect. This form of mathematical domino-toppling allows an infinite number of cases to be proved by just proving the first one.
Appendix 10
shows how proof by induction can be used to prove a relatively simple mathematical statement about all numbers.

The challenge for Wiles was to construct an inductive argument which showed that each of the infinity of elliptic equations could be matched to each of the infinity of modular forms. Somehow he had to break the proof down into an infinite number of individual cases and then prove the first case. Next, he had to demonstrate that, having proved the first case, all the others would topple. Eventually he discovered the first step to his inductive proof hidden in the work of a tragic genius from nineteenth-century France.

Evariste Galois was born in Bourg-la-Reine, a small village just south of Paris, on 25 October 1811, just twenty-two years after the French Revolution. Napoleon Bonaparte was at the height of his powers, but the following year saw the disastrous Russian campaign, and in 1814 he was driven into exile and replaced by King Louis XVIII. In 1815 Napoleon escaped from Elba, entered Paris and reclaimed power but within a hundred days he was defeated at Waterloo and forced to abdicate once again in favour of Louis XVIII. Galois, like Sophie Germain, grew up during a period of immense upheaval, but whereas Germain shut herself away from the turmoils of the French Revolution and concentrated on mathematics, Galois repeatedly found himself at the centre of political controversy, which not only distracted him from a brilliant academic career, but also led to his untimely death.

In addition to the general unrest which impinged on everybody's life, Galois's interest in politics was inspired by his father, Nicolas-Gabriel Galois. When Evariste was just four years old his father was elected mayor of Bourg-la-Reine. This was during Napoleon's triumphant return to power, a period when his father's strong liberal values were in keeping with the mood of the nation. Nicolas-Gabriel Galois was a cultured and gracious man and during his early years as mayor he gained respect throughout the community, so even when Louis XVIII returned to the throne he retained his elected position. Outside of politics, his main interest seems to have been the composition of witty rhymes, which he would read at town meetings to the delight of his constituents. Many years later this charming talent for epigrams would lead to his downfall.

At the age of twelve Evariste Galois attended his first school, the Lycée of Louis-le-Grand, a prestigious but authoritarian institution. To begin with he did not encounter any courses in mathematics and his academic record was respectable but not outstanding. However, one event occurred during his first term which would influence the course of his life. The Lycée had
previously been a Jesuit school and rumours began to circulate suggesting that it was about to be returned to the authority of the priests. During this period there was a continual struggle between republicans and monarchists to sway the balance of power between Louis XVIII and the people's representatives, and the increasing influence of the priests was seen as an indication of a shift away from the people and towards the King. The students of the Lycée, who in the main had republican sympathies, planned a rebellion but the director of the school, Monsieur Berthod, uncovered the plot and immediately expelled the dozen or so ringleaders. The following day when Berthod demanded a demonstration of allegiance from the remaining senior scholars, they refused to drink a toast to Louis XVIII, whereupon another hundred students were expelled. Galois was too young to be involved in the failed rebellion and so remained at the Lycée. Nevertheless, watching his fellow students being humiliated in this way only served to inflame his republican tendencies.

It was not until the age of sixteen that Galois enrolled in his first mathematics class, a course which would, in the eyes of his teachers, transform him from a conscientious pupil into an unruly student. His school reports show that he neglected all his other subjects and concentrated solely on his new found passion:

This student works only in the highest realms of mathematics. The mathematical madness dominates this boy. I think it would be best for him if his parents would allow him to study nothing but this. Otherwise he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.

Galois's desire for mathematics soon outstripped the capacity of his teacher, and so he learnt directly from the very latest books written by the masters of the age. He readily absorbed the most
complex of concepts, and by the time he was seventeen he published his first paper in the
Annales de Gergonne.
The path ahead seemed clear for the prodigy, except that his own sheer brilliance was to provide the greatest obstacle to his progress. Although he obviously knew more than enough mathematics to pass the Lycée's examinations, Galois's solutions were often so innovative and sophisticated that his examiners failed to appreciate them. To make matters worse Galois would perform so many calculations in his head that he would not bother to outline clearly his argument on paper, leaving the inadequate examiners even more perplexed and frustrated.

The young genius did not help the situation by having a quick temper and a rashness which did not endear him to his tutors or anybody else who crossed his path. When Galois applied to the Ecole Polytechnique, the most prestigious college in the land, his abruptness and lack of explanation in the oral examination meant that he was refused admission. Galois was desperate to attend the Polytechnique, not just because of its academic excellence but also because of its reputation for being a centre for republican activism. One year later he reapplied and once again his logical leaps in the oral examination only served to confuse his examiner, Monsieur Dinet. Sensing that he was about to be failed for a second time and frustrated that his brilliance was not being recognised, Galois lost his temper and threw a blackboard rubber at Dinet, scoring a direct hit. Galois was never to return to the hallowed halls of the Polytechnique.

Undaunted by the rejections, Galois remained confident of his mathematical talent and continued his own private researches. His main interest concerned finding solutions to equations, such as quadratic equations. Quadratic equations have the form

The challenge is to find the values of
x
for which the quadratic equation holds true. Rather than relying on trial and error mathematicians would prefer to have a recipe for finding solutions, and fortunately such a recipe exists:

Simply by substituting the values for
a
,
b
and
c
into the above recipe one can calculate the correct values for
x
. For instance, we can apply the recipe to solve the following equation:

By putting the values of
a
,
b
and
c
into the recipe, the solution turns out to be
x
= 1 or
x
= 2.

The quadratic is a type of equation within a much larger class of equations known as polynomials. A more complicated type of polynomial is the cubic equation:

The extra complication comes from the additional term
x
³
. By adding one more term
x
⁴
, we get the next level of polynomial equation, known as the quartic: