Fermat's Enigma

by Simon Singh

Then, on March 1, 1847,

Gabriel Lamé, who had proved the case n = 7 some years earlier, took the podium in front of the most eminent mathematicians of the age and proclaimed that he was on the verge of proving Fermat’s Last Theorem.

as soon as Lamé left the floor Augustin Louis Cauchy, another of Paris’s finest mathematicians, asked for permission to speak. Cauchy announced to the Academy that he had been working along similar lines to Lamé, and that he too was about to publish a complete proof.

three weeks after they had made their announcements they deposited sealed envelopes at the Academy.

If a dispute should later arise regarding the originality of ideas, then a sealed envelope would provide the evidence needed to establish priority.

many of them secretly hoped that it would be Lamé and not Cauchy who would win the race. By all accounts Cauchy was a self-righteous creature, a religious bigot and extremely unpopular with his colleagues. He was tolerated at the Academy only because of his brilliance.

It was neither Cauchy nor Lamé who addressed the Academy but rather Joseph Liouville. Liouville shocked the entire audience by reading out the contents of a letter from the German mathematician Ernst

In parallel with his military career Kummer actively pursued pure mathematical research

To Kummer it was obvious that the two Frenchmen were heading toward the same logical dead end, and he outlined his reasons in the letter that he sent to Liouville.

According to Kummer the fundamental problem was that the proofs of both Cauchy and Lamé relied on using a property of numbers known as unique factorization.

Unique factorization was discovered back in the fourth century B.C. by Euclid,

Unfortunately both of their proofs involved imaginary numbers.

Kummer pointed out that it might not necessarily hold true

severely damaged the proofs of Cauchy and Lamé, but it did not necessarily destroy them completely.

These so-called irregular primes, which are sprinkled throughout the remaining prime numbers, were now the stumbling block to a complete proof.

Lamé realized that had he been more open about his work he might have spotted the error sooner,

Cauchy refused to accept defeat. He felt that compared to Lamé’s proof his own approach was less reliant on unique factorization,

For several weeks he continued to publish articles on the subject, but by the end of the summer he too fell silent.

Kummer had demonstrated that a complete proof of Fermat’s Last Theorem was beyond the current mathematical approaches.

The situation was summarized by Cauchy,

after many times being put forward for a prize, the question remains at the point where Monsieur Kummer left it.

specially by Monsieur Kummer;

it would adjugate the medal to Monsieur Kummer, for his beautiful researches on the complex numbers composed of roots of unity and integers.

And in some of those cases the flash of insight that solved the problem did not rely on new mathematics; rather, it was a proof that could have been done long ago.

Having learned all there was to learn about the mathematics of the nineteenth century, Wiles decided to arm himself with techniques of the twentieth century.

The Wolfskehl family were famous for their wealth and their patronage of the arts and sciences, and Paul was no exception. He had studied mathematics at university

The story begins with Wolfskehl’s obsession with a beautiful woman, whose identity has never been established.

he decided to commit suicide.

He set a date for his suicide and would shoot himself through the head at the stroke of midnight. In the days that remained he settled all his outstanding business affairs, and on the final day he wrote his will and composed letters to all his close friends and family. Wolfskehl had been so efficient that everything was completed slightly ahead of his midnight deadline, so to while away the hours he went to the library

Wolfskehl worked through the calculation line by line. Suddenly he was startled at what appeared to be a gap in the logic—Kummer had made an assumption and failed to justify a step in his argument.

Wolfskehl was so proud that he had discovered and corrected a gap in the work of the great Ernst Kummer that his despair and sorrow evaporated. Mathematics had renewed his desire for life.

Paul had bequeathed a large proportion of his fortune as a prize to be awarded to whomsoever could prove Fermat’s Last Theorem.

Sam Loyd Senior. Loyd’s most famous creation was the Victorian equivalent of the Rubik’s Cube, the “14–15” puzzle, which is still found in toy shops today.

Loyd’s puzzle and the disorder parameter demonstrate the power of an invariant. Invariants provide mathematicians with an important strategy to prove that it is impossible to transform one object into another.

the study of knots, and naturally knot theorists are interested in trying to prove whether or not one knot can be transformed into another by twisting and looping but without cutting.

The concept of an invariant property is central to many other mathematical proofs and, as we shall see in Chapter 5, it would be crucial in bringing Fermat’s Last Theorem back into the mainstream of mathematics.

Appendix 6 shows the sort of classic error that can easily be overlooked by an enthusiastic amateur.

mathematicians began to examine the foundations of their subject in order to address some of the most fundamental questions about numbers.

greatest figures of the twentieth century, including David Hilbert and Kurt Gödel, tried to understand the most profound properties of numbers in order to grasp their true meaning and to discover what questions number theory can and, more important, cannot answer.