Wednesday, December 24, 2008

Book Summary: Fermat's Enigma by Simon Singh

One of the best books I’ve read in a while. This book should be read like a detective story that spans centuries, tiny nuggets of information collected all the way till the 20th century when the last greatest theorem ever posed was solved: Fermat’s Last Theorem.

For those not in the know, Fermat’s Last Theorem was, for 300-odd years, a conjecture. To get an idea about it, we begin with the familiar Pythagoras theorem: x^2 + y^2 = z^2
Now, it is possible to get an infinite number of combinations of whole numbers x, y and z that satisfy this equation. Whole numbers are counting numbers like 1,2,3, etc. and do not include fractions or negative numbers. On the other hand, the genius that is Pierre de Fermat discovered in 1637 that if you increase the power even by one, there are NO whole number solutions. The statement goes:

The equation x^n + y^n=z^n has NO whole number solutions for any whole number n higher than 2.

Looks simple, doesn’t it? But the greatest mathematicians of the last 300 years have been unable to provide a complete proof for that statement till Andrew Wiles came along. Slowly but surely, the mathematicians over the years have been chipping away at the conjecture, but it was Wiles’ magnum opus in 1995 which finally settled the issue once and for all. Here is a short approximately chronological list of the important events leading to the proof of the last greatest mathematical riddle.

In 1637, while reading Book II of Claude Gaspar Bachet’s Latin translation of the medieval mathematical treatise Arithmetica (by Diophantus), Pierre de Fermat made a marginal note near the Pythagoras Theorem. He mentions that increasing the power to any whole number greater than two results in the number of (whole number) solutions going from infinitely many to zero. Also, he mentions discovering a marvelous proof of the same, but no space to write it down in the margin. This statement was to tease, tantalize, torment and torture mathematicians for the next three centuries.

In another context, Fermat specifically proves the case for n=4 using the method of infinite descent. Note that the equation is structured such that proving it for all prime numbers (indivisible numbers like 2,3,5,7,13…) is alone sufficient to proveit for all numbers. If we prove for n=3, that automatically proves it for n=6,9,12,... The mathematical genius Leonhard Euler used the concept of imaginary numbers (built on i, the imaginary square root of -1) to prove the case n=3, but was unable to extend it or generalize it to higher primes. Sophie Germain (one of the few women mathematicians who rose beyond petty circumstances to greatness) showed that for special prime numbers (called Sophie Germain primes), even if Fermat’s conjecture was false, then there exists an extremely tight restriction that makes a solution extremely unlikely (but not entirely impossible).

Based on work by Sophie Germain, Gustav Lejeune-Dirichlet and Adrien-Marie Legendre independently proved Fermat’s theorem for n=5. Later, Gabriel Lamé extended it for n=7. Lamé and Augustin Louis Cauchy had a very public rivalry (working on similar lines) to arrive at the generalized proof when their hopes were dashed by Ernst Kummer who showed that their lines of thinking had a fatal flaw: they assumed unique factorization (i.e. every number can be made by multiplying a unique number of primes, e.g. 6 = 2x3 and 9 = 3x3), which doesn’t hold for complex numbers. A flaw in Kummer’s reasoning was then fixed by an amateur mathematician Paul Wolfskehl (who almost committed suicide, but then got interested in the theorem) making the proof for Fermat’s Last Theorem even more unattainable.

As an aside, spearheaded by David Hilbert, mathematicians were re-examining and re-establishing every mathematical statement ever made using the most basic of mathematical axioms (assumptions such as 1+1=2). Kurt Gödel dealt a death-blow to this effort by proving that given a set of consistent axioms, it is possible to make statements that can never be proved true or false. Worse still, given a set of axioms, we cannot even prove that they are consistent. This finding caused a lot of disillusionment, especially with unsolved problems like Fermat’s Last Theorem, opening up the possibility that theoretically, we may never be able to find an answer and that Fermat may have made a mistake in reasoning.

With the invention and rapid development of computers, brute-force methods were employed to search for solutions that disprove Fermat’s Last Theorem. None were found, but that cannot remove the niggle of doubt that mathematicians subsist upon. Mathematicians know for a fact that searching an infinite space, even to a significant extent, can never give the right answer. For example, 31, 331,3331,33331 are all primes. However, the extrapolation is destroyed by the fact that 333333331 is not a prime number. Also, a conjecture similar to that of Fermat was made by Euler (no whole number solutions exist for x^4 + y^4 +z^4 = w^4) and was proven wrong using brute-force search by Noam Elkies.

In the modern era, Fermat’s Last Theorem was almost forgotten and was just an ancient romantic mathematical curiosity. Developments in two other fields were soon to bring the theorem under the active scrutiny of mathematicians.

One of these fields was the branch of elliptical equations (which take the form y^2 = x^3 +ax^2+bx+c) which are hopelessly complicated if you attempt to find solutions. So a technique called clock arithmetic was used on them. Clock arithmetic takes a small subset of the number line and loops it onto itself, much like the clocks we use to measure time. Our standard clocks and watches are 12-clocks, i.e. 12=0 and the arithmetic changes accordingly. For example, 2 hours after 11am is 1pm, i.e. 11+2=1. If we take a series of clocks (1-clock, 2-clock, 3-clock, etc.) and see how these elliptical equations look like in clock arithmetic, we see that they have different number of solutions in different clocks. For each elliptical equation, we can write down an infinite sequence of numbers which show the number of solutions in each clock. This sequence is called the characteristic L-series of the elliptical equation.

The other field was modular forms, which are exotic mathematical entities living in a four-dimensional space. Just like complex numbers can be constructed with one real number and one imaginary number, a modular form is constructed from numbers from 4 different dimensions. Modular forms are similarly decomposed as containing an infinite number of components. Worse still, they can contain different numbers or “quantities” of each component. Again, these numbers can be arranged in an infinite sequence called the characteristic M-series of the modular form.

Around 1955, Yutaka Taniyama and Goro Shimura floated around a hypothesis forever linking the two fields: Every elliptical equation with its L-series has a corresponding modular form with an identical M-series, and vice-versa. This was an exceedingly bold statement and was called the Taniyama-Shimura Conjecture. Every elliptical equation form that was examined and scrutinized verified the conjecture. However, examples don’t maketh a proof and the hypothesis remained a conjecture, with no mathematical proof. However, it formed the starting point of a new generation of mathematicians, taking number theory to new heights, but nevertheless with a sense of discomfort that it was still a conjecture.

Andrew Wiles who had been fascinated by Fermat’s Last Theorem as a child, decided to temporarily abandon his fascination in favor of more ‘productive’ math and joined as a Ph.D. student under the supervision of John Coates in 1975. Coates encouraged him to study elliptical equations and soon, he became an expert on them. His childhood dream of proving the Last Theorem was soon to be resuscitated, though. In 1984, Gerhard Frey announced a flashing insight that if Fermat’s Theorem was actually false, then any solution for that equation can be “rearranged” to give an elliptical equation. Surprisingly, the L-series of this equation seemed to be highly unlikely as an M-series of a modular form. Ken Ribet later redefined ‘unlikely’ as ‘impossible’. These developments carried a startling inference: the Taniyama-Shimura Conjecture implied Fermat’s Last Theorem. If one was true, then the other had to be.

Wiles realized that he was now actually equipped to deal with his childhood dream. All he needed to do was unequivocally prove the Taniyama-Shimura Conjecture. He decided to work in seclusion till he mastered his prize. It took him seven years spent in mastering modern mathematical techniques and applying, modifying, redefining and strengthening them to suit his case.

Wiles decided to use a time-honored technique called mathematical induction to prove Fermat’s Theorem. Given an infinite sequence of statements to prove, the principle of mathematical induction says that it is enough to prove two things: one, the first statement is true; two, if one statement is true, then the next statement is automatically true. This is the mathematical version of the domino effect: if dominoes are arranged next to each other in a sequence, toppling the first one will topple all of them in sequence.

The problem Wiles faced was that not only a single L-series have infinite elements, but also, there are an infinite number of L-series as well. This was infinity piled upon infinity, in other words, an infinite number of domino lines, each of which stretched to infinity. Using a technique called group theory (developed by the 19th century tragic genius Évariste Galois), Wiles was, able to ‘sort’ the infinite number of domino-lines (read, elliptical equations) into a bunch of groups. Painstakingly, he also proved the Taniyama-Shimura Conjecture for the first element of each group. All that was required now was to show that within each group, toppling one domino toppled the rest.

Using a methodology called Iwasawa theory, which he learnt under John Coates, Wiles attempted to show the domino effect for each group. However, after a couple of years of experimentation, Wiles gave up on Iwasawa theory and looked around for something else. Another student of Coates, named Mattheus Flach worked on extending a previous method by Victor Kolyvagin in the study of elliptical equations. Wiles found renewed hope in the Kolyvagin-Flach method and worked on extending it to deal with his groups, but found the mathematics out of his expertise. Using help from Nick Katz, he slowly and steadily applied the difficult method and proved the domino-effect for each of his groups.

In a series of lectures in June 1993, Andrew Wiles finally delivered his proof to the audience of mathematicians. Flying rumors about the Taniyama-Shimura Conjecture had ensured that most renowned mathematicians had arrived for the lecture, just in case they were true. To a packed audience of the best mathematicians of the present decade, Wiles delivered what was to be known as the Lecture of the Century. He outlined his proof of the Taniyama-Shimura Conjecture using Galois representations and the modified Kolyvagin-Flach method. By inference, this was the proof of Fermat’s Last Theorem. At the end of it all, he wrote out the statement of Fermat’s Last Theorem and said “I think I’ll stop here.”

Among cheers and standing ovations, mathematicians let out a collective sigh. An ancient riddle had been cracked, but a beloved and inspirational puzzle had been lost. Such is the way of math. Fermat’s challenge had been finally met.

Outlining a proof on the blackboard does not substitute for detailed scrutiny of the written proof by your peers. Soon enough, manuscripts containing the detailed proof were submitted and sent to referees for checking. Nick Katz, one of the referees, soon spotted an error. Wiles believed he had modified the Kolyvagin-Flach method to be sufficiently strong enough to deal with his groups. Unfortunately, the expert in Katz showed it to be otherwise. This left a hole in the proof which had to be filled before the proof can be agreed to be true. Wiles believed that this would be a minor correction. However, it proved to be a stumbling block and the correction never materialized, no matter how hard he tried. Wiles became depressed and dejected and wondered if seven years of work were going to go down on a technicality or whether he had been barking up the wrong tree all the while.

Finally, after struggling hard with his inner demons, Wiles had his final flash of insight that would rest the problem forever. He had previously discarded Iwasawa theory as inadequate. Similarly, the Kolyvagin-Flach method had also turned out to be inadequate. But Wiles realized that putting the two together, the combination was irresistible. In a matter of weeks, Wiles had combined the two methods to publish irrefutable proof for the Taniyama-Shimura Conjecture and Fermat’s Last Theorem. The new manuscript was even more streamlined and cohesive than the previous one.

Fermat’s Last Theorem was finally laid to rest in the minds of the geniuses of this century.

A couple of words about the book itself. Simon Singh has done an excellent job in compiling the history of mathematics. The book is a history book disguised as a detective story. A must-own for any math lover. Singh relates events from centuries past and present that will inspire you to take up mathematics today. Especially riveting are the tragic stories about Galois and Taniyama. The historical context adds a great deal to the story and increases one’s appreciation for the fact that the things we trivially ‘assume’ today have actually come from a lot of work by geniuses of the past.

1 comment:

Shantanu Shankar Bhattacharyya said...

Awesome....I loved your post buddy!!