Monday, July 30, 2012

Fermat's Enigma - Simon Singh

"I think I'll stop here ", concluded Prof Andrew Wiles , his third and final lecture of the series at the Cambridge University, to the transfixed audiences which included some of the prominent mathematician of the world. Only a few understood those symbols and equations that was scribbled on the board. But most of them knew that they witnessed one of the biggest moment in the history of mathematics. The 300 year old, longest standing mathematical problem has now been solved. Fermat's Last Theorem, which puzzled, frustrated and challenged the brilliant minds across the globe for over three centuries is now been addressed. The Misery resolved, the mathematicians around the world is now rejoiced.

Some of the toughest problems of the world are simple, yet difficult to prove. We all have learned the Pythagoras' theorem of right angles triangles.  a (2) + b(2) =c(2) where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

Mathematician who followed, Pythagoras , like Euclid , Dyophantus and the those of Hypatian school of Alexandria made early progress in the development of Mathematical Sciences. During renaissance period, the newly learnt wisdom of Arab's and the Eastern world, the European thinkers and mathematicians gave giant leap to the science of mathematics.

In the early 17th century Frenchman Pierre de Fermat, an amateur mathematician was famous for his riddles, usually aimed at those across the Channel at England. While exploring the Dyaphantus book 'Arithametica' he wrote his famous observation which later came to be known as 'Fermat's Last Theorem' , tormenting the mathematicians around the world for next three centuries. His theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. On the margins of the book he also noted that "I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain." 
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. ( it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain)
The master riddler, now sent the world in a spin with rather innocent looking observation. The next few centuries saw mathematicians trying to find out the elusive proof, which Fermat supposed to have found but did not bother to write down for want of space in the margin ! The "number theory " branch of mathematics saw quantum progress, thanks to one riddler called Pierre de Fermat.

The world had been seeing many such conjectures and theorems over the years. Many were either proven or broken with the latest methods or the technological improvements, largely with the invention of computers. However, this one remained elusive. Some people even gave up trying, despite some large amount of money being promised by many institutes and individuals.

Simon Singh's book on the triumph over Fermat's theorem , takes us through this mathematical journey. From the early days of Mathematics, secretly hidden in the "Pythagoran sects" later flourished under the various rulers and thinkers in the early ages, to the days of Fermat and the later mathematicians, directly of indirectly influencing the progress in number theory. This book, resulted of a documentary he directed for BBC on Andrew Wiles ( see the you tube link here) , is a detailed exploration of the birth and its progress through the millennium, the centuries, decades and years of progress of mathematical wisdom.

There were tricksters as well. People who churn out 'theorems' and conjectures to the world, And there are many which at the first look appears to be true. Look at the below sequence.

31 ; 331 ; 3,331 ; 33,331 ; 333,331 ; 3,333,331 ; 33,333,331 ...all these are prime numbers. However, the next number in this pattern 333,333,331 is not a prime. 333,333,331 = 17 x 19,607,843.

Similarly, Euler , 18th century mathematician extrapolated the Fermat's Theorem with his on version. Euler's conjecture said , there are no whole number solution for x 4 + y4 + z4 =w4. For two hundred years nobody could prove Euler's conjecture.On the other hand , nobody could disprove it by finding a counter example. LAck of counter example is a strong evidence in favour of the conjecture. Then, in 1968, Naom Elkies of Harward University discovered the following solution

2682449(4) + 15365,639(4) + 18,796,760(4) = 20,615,673(4)  , thus proving Euler's conjecture wrong..

Various mathematical wiz-kids attempted this theorem, moving towards the proof. In the 18th and 19th century, the proof for the power of 3,4,5 and upto 7 were established. But none of them is not good enough in the strict mathematical arena as the proof for the theorem. With the help of the computers and the progress over the centuries, mathematicians now are able to use the new techniques and tools that are available to them. Improvements in the other areas of mathematics, especially in the algebraic equation helped advancement in this regard. Then in 1959, two Japanese Mathematicians, made a great announcement comparing 'elliptical equations' to the 'modular forms', which later came to fame as Taniyama-Shimura Conjecture. There were attempts to prove this and a theorem comparing Taniyama-Shimura conjecture to the Fermat's Last Theorem was evidently established. Which in other words mean that if Taniyama-Shiumura Conjecture is proved, Fermat's Last Theorem is true. With the renewed vigour, mathematicians around the world got back to their scratch pads, trying to get the conjecture proved.

Andrew Wiles, by now a professor at Princeton , had his childhood dream on working and proving Fermat's theorem. The rest of the book is on his 7 years of efforts , his small achievements and many set backs on working on his dream project. to protect his efforts and achievements ( you do the 99% of the works and taking it from here someone else completing the job, thus taking all the claim is something he wanted to avoid) he worked in secrecy, even hiding from his own close associates and friends.

While Andrew Wiles was shut himself in a room working with single objective, a Japanese Mathematician Miyaoka in 1988 announced to the world that he has found a proof, shattering Wiles dreams. However, to the relief of Andrew, his claim was also failed to break the long pending wait for proof , triggering an interesting graffiti at the NY underground with the artist declaring " I have discovered a truly remarkable proof of this, but I can't write it now because my train is coming."

Then on the eventful day 23 June 1993, he announced to the world at his alma matar , in the presence of his mentor , that he had found the proof to the Fermat's Last Theorem. But the process of verification and authentication remain. The journal has to recruit some of the best judges to go through his manuscript and scrutinize every line to make sure there are no mistakes, any points overlooked or made any un-clarified assumptions. As feared, there was a glitch as one of the points did not work in all the possible hypothesis. To cut short, the next one and half years were heartburn for Andrew. As is in many cases, he was about to abandon his effort and accept the inevitable, the solution to the problem appeared to him. He was more relieved than elated.

Now that the long standing problem is solved, the mathematical world is suddenly has no new challenge (well, not to this level of complexity). That also means, the world need another motivation that can trigger the next wave of progress. As Wiles mentioned, the sense of purpose is now lost. The remaining of the days will not be as directed as it used to be. There is still hope for the enthusiasts. Most of the methods and tools by Andrew Wiles and his contemporaries were not available to Fermat while he worked on this. And if his claim is true, then there is a much simpler way to prove the theorem. The quest is now on to find out the proof Fermat had in his mind..
Fermat's Enigma ( 1997)

Simon Singh

Anchor Books

315 Pages
NY Times, Leegruenfield,

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