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..
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Fermat's Enigma ( 1997)

Simon Singh

Anchor Books

315 Pages
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NY Times, Leegruenfield,

Underworld - Don DeLillo

The day Bobby Johnson hit a home run to win the National League title for Giants against Dodgers, USSR successfully detonated the second Nuclear Bomb. Don DeLillo's master piece starts at this moment, with the match description and the boy who managed to collect the ball that went into the crowd, which his alcoholic father sells to an enthusiast for 23 Dollars.

The united states in the second half 20th century, the war is won and after the nuclear bomb is dropped in Japan. The imminent threat of USSR are looming in the air, the world is getting divided between the Socialists and Capitalist economies. USSR is building capabilities to compete with the mighty US. The explosion of the Nuclear Bomb is one of their signal to the other world of their intention. Cold war between the nations are at the peak. The Cuban Missile Crisis, the McCarthy era, nuclear proliferation and the rest of the events that was prominent in the 50 years since the World War. The common lives in the US is deeply influenced by the effects of cold war. The anxiety, the fear resulting in various precautionary measures. Children are brought up with emergency measures. These realities of the new world is the back drop of DeLillo's novel.

However, DeLillo construct his mammoth book around Nick Shay, a garbage and waste collection and disposal expert ( who incidentally now owns the baseball that was hit for the home run in 1951. Nick with a past of an accidental murder, being in juvenile retention center before beginning his new life at a Jesuit Reform School. Most of the events revolve around Nick and his immediate surroundings, his wife who harbors an illicit relation with his close friend, his brother, other friends and colleagues. Structurally told in reverse chronological order fro the 90s to the 50s through it various incidents rebuilding the life during cold war.

With these loosely held narrative, he builds the modern American society under the influence of the economic and political scene. However, this book is not much of a plot, a character development or a story. Its about writing. May be that is why the experience is in reading. Its too intelligent to the reader. Its is too perfect from the construction and the formation of words , sentences and paragraphs. In its effect to retain the medium at its prime, the rest are not given much importance. The world continue to discuss this as a literary phenomenon ( as the greatest American Novel of 20th century and all that) is precisely comes from this ambiguity that he build around his own world. The controlling hand of the writer is omnipresent often hindering the smooth reading. Which is why even after reading this mammoth book, one can not warm up to it.

The reaction within me was contradictory. Glorious at times, Boring and pretentious at places, engaging and interesting at places, yet dull and dragging at other times. Its massive, by size and accomplishment. Its definitely not an easy read and not easy to fully comprehend. I think he manages to bring out the frustration, fear, anxiety and related major events during the cold war years. At the end, its a sense of relief after you are through with 830 pages.

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Underworld ( 1997 )

Don DeLillo

Picador

827 pages
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Wiki Entry, Guardian, Review and Responses, The New Canon

Chowringhee - Sankar

Few novels have the ability to captivate you. There are no major twist or turns like a thriller, the story is simple and normal, yet absorbing. The book could be noted as dated, for it is written and published in 1962, yet it has an aura of freshness around it. Set in the early 50s, in a Metropolitan hotel in Calcutta ( named Hotel Shahjahan ), this often anecdotal, mostly incident based narrative revolving around the hotel employees and the guests takes us to an entirely different world of existence. The worlds war II is ended and the Britishers are folded back to their country leaving India to the control of Indians. However, many Victorian lifestyle remain with the urban elite of India, with their evening parties and elite gathering.

Told in first person narrative by Shankar , who started his career under Calcuttan barrister Noel Barwell ( an overlap with the writer Sankar can be observed here) after his father passed away, before being employed at Hotel Shahjahan which boast its Victorian heritage. Young Sankar, a novice join the work group at the Shahjahan Hotel as a reception help, being recommended by his friend Byron an Anglo-Indian private detective. Under the fold of experienced and dashing Satyasundar Bose ( Sata Bose as he is called) , Shankar learns the primary lessons in managing the hotel. The functioning of various teams, the men behind these tasks and their own life beyond what is exhibited to the guests with a smile. Shankar realises the dual world that exists within the hotel and outside on every character he meet , be it the guests , the businessmen who frequent, the hotel staffs or the manager himself, to which he is a silent witness and often made to participate . Beyond the facade of the building, has a life which are hard and often sad. The manager Marco Polo, an Italian who moved in to take charge from a reputed hotel in Rangoon had his own past connected to the city. A story of love and treachery, so is the 'dhobi', a brahmin lament over his ill-luck of having to wash dirty linen. His mentor Sata Bose , the stenographer Rosie, the musician Gomez, various dancers who frequent Shahjahan to perform Cabaret are treasure trove of stories and experiences to tell.

It is this duality of life, I guess, makes this novel interesting to read. Forced to present themselves to the world demanded by their profession and an inner world which is suppressed to the small world of their own existence. Shankar does a marvellous job in getting these dual world to the best effect with sublime use of language and emotion. While the story moves through a predictable outcome at the end, the importance is not for the plot per se, but for the narration and narrative space. Shankar , writes some moving episodes of the lives of few individuals, in its richness and depth. One Dr.Sutherland, who comes in search of his ancestors, The Scottish Dancer and her dwarf brother, the Air hostess who falls in love with Sata Bose, the musician Gomez and his 17th century Mozzart, Brahms and other records, Rosie and her failed marriage, the hostess at the Suite number one employed to entertain the corporate guests for a business family who committed suicide, the filmstar who take refuge at the hotel running away from her husband... Sankar gives some memorable and characters to the readers,

The beauty of the book is in its mosaical narrative. The kaleidoscopic images of the interconnected members of the hotel and the guests. Those with long lasting involvement and interests in the hotel. Microcosm of the world revealed within the four walls of the empire of Shahjahan hotel. As one character says "every brick in this Hotel has a novel in it".

Interestingly, this book was written and published three years before Aurthur Haileys celebrated Hotel, which went on to become a best seller book as well as movie. Aurthur Hailey's book deals with the happening around the hotel and not much to do with the inner lives of its employees and its guests. I haven't read Joseph Roth's "Hotel Savoy", which I understand is a different class altogether. To me, despite its lack of wholesomeness, this book has some beautiful writing, remarkably translated by Arunava Sinha.

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Chowringhee ( 1962 )

Sankar ( Mani Sankar Mukherji) - Translated from Bengali by Arunava Sinha ( 2007)

Penguin Books

403 Pages
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Guardian, the Economics, Behind the scene, Neel Mukherjee

ലൈവ് ഗ്രീന്‍ - 30 (Live Green 30) - Ajith Janardanan

Few years ago, Jose Saramago published a book called "Death at Intervals". Building up a world of chaos at the result of death's absence, in his own inimitable style. Satirical , witty and with great restraint ( its easy to go haywire with such a theme), he produced a brilliant work to me the last significant work by the Nobel Laureate. When I read the blurb of Ajith Janardanan's new book, the reference of Saramago's book came into mind. I am not comparing a novice to the Nobel Laureate, but bringing to your attention a class work around death, or its absence there of.

Ajith Janardananan's short Novel, which won the Manorama Online Novel Carnival award ( I guess it can be read online here), looks at the duality of world, where the mortals, won over the death thus eventually achieving what the entire human civilisation was working towards for generations to generations. They discovered the magic medicine, to prevent death and aging. However, for every step towards progress(?), there have to be an opposition. There were equally strong rejection to the medicine ( called Live Green 30) and defying the will of God by the other half of populace. The world debated and split in line with their believes. Those who supported 'immortality" chose to find their own world called "Bravo" and those who wish to remain faithful to the God's wishes retained their world 'Alpha'. A Great migration between the worlds ensured clear differentiation despite the initial chaos and loss of human lives.

For the people of Bravo, the initial excitement did not last long. The population grew in the absence of death. The had to mandate that people take necessary precaution ( giving up birth) before they have been accepted to the fold of 'immortals' by ceremoniously taking the 'Live Green 30'. There were rebellious voices which had to be curbed. The inventor himself ( One David Kinman) came out against his invention ( albeit discreetly). Many people grew disappointed with life and wanted to return to the normalcy. There are no new births of kids and many professions seems to be of no value in the new system ( gynecologist, teachers).

On the other side of the world, David Kinman's son ( who along with his mother refused to move to the new world) trying to find remedy to his father's mistake. He believed it is his moral obligation to correct his father's error. While the youngsters are now finding the 'immortality' an attractive options and certain quarters are rumored to have been developing the clandestine version of Live Green 30. There were also news of an imminent attack by the Bravo , struggling to provide for its growing population with resources.

It is in this world where few youngsters in Bravo, attempting to make some changes. They build temples, "old age homes" for people disillusioned with life, maternity clinic for those refuse to tow the line of authorities. They also wanted David Kinman to break his silence and talk to the world, by attempting to get an interview published. The book then looks at the various possibilities for both Alpha and Bravo voices of rebel and authorities , not very convincingly though.

While the short novella has nothing to claim in the literary side, it is a curious attempt to build upon a possibility in its fullest extend. To me, Ajith Janardanan had succeeded in that front. Despite few clieched, often expected lines of writing, it did bring out a freshness in the narration in Malayalam writing.

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ലൈവ് ഗ്രീന്‍ - 30 / Live Green 30 ( 2012)

Ajith Janardanan

D C Books

79 Pages

Rs 55
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A Sunday at the Pool in Kigali - Gil Courtemanche

".. talked without stopping, about AIDS, about corruption and about massacres. He repeated what (he) had said a thousand times before...."

The story of every genocide takes the same often narrated path of violence and massacre. The second World War, the balkan conflict or what the world witnessed in Africa. The story of molestation, violence, rape and massacre repeat itself, by merely changing the country or continent. Canadian journalist and writer, Gil courtemanche, brings the world attention again to the gruesome history of the 1994 Rwandan genocide with a powerful, violent and often angry portrayal of the war time Kigali. The writing is hard hitting, it is upsetting and he uses extreme graphic images be its the effect of AIDS, the massacre or the rape of women.

Hotel des Mille-Collins, at the centre of Kigali is the nerve-centre of the actions in Rwanda. Apart from hosting the UN peace keeping soldiers, it is also the point of action for various Global organisations, working in Rwanda, like the UNICEF, Red Cross, French and Belgian embassies, the rich expatriates and the high class prostitutes. Bernard Valcourt, a Canadian journalist ( an alter ego of the writer), comes to Kigali to set up the television station in Rwanda with the help of Canada. Widowed, and his daughter now married and settled, he accepted the assignment to escape his boring life. The telecast project is perpetually delayed, and he and his crew are now opened to the world of another epidemic that is creating havoc in Africa - AIDS. Focussing his camera and his journalistic intuition, he and his team set about filming some of the victims and their last days. Though he made several friends amongst the victims and was able to film some of the touching scenes of their last breath, his documentary did not receive the attention neither in Rwanda, nor to the rest of the world.

Death and sex are two companions in the street. Scores are dying of AIDS, and the rest are victims of the violence. Machette clad predators on prowl for their victims, so are the prostitutes carrying deadly HIV virus.

The build up for a potential show down between Tutsis and Hutus were already evident. There are sporadic incidents of violence and arson. With most of his friends belongs to the minority Tutsi community, his attempt to bring justice to those affected were received with no enthusiasm from the ruling majority, and he was often warned with dire consequences. His diplomatic immunity , being a Canadian did not help him, and his attempts to get the international attention to the crisis, about to explode had no success, barring a small publication based out of Belgium. As one after other of his friends being killed by a meticulous planned genocide by the Hutu majority ( a plan very similar to what we had seen in the extermination of Jews in the first half of last century), Valcourt continue to work on his capacity.

The crisis blow up with a full scale action of murder and mass migration of Tutsis after a political crisis resulted in the murder of the President, the country goes into chaos. Anarchy rules and the foreigners are rescued to the nearby countries. Valcourt, now in love with a local girl Gentille, a Hutu with the physical features of Tutsi, refuses to leave and plans for his wedding. The Rwandan crisis now turns into a war after the Tutsi rebel regroup (with the help of Uganda) and return the favour. Now the exodus is from the Hutu quarters and no one is safe in the streets any more. In an attempt to flea , Valcourt and his wife was stopped and she was detained for not having the valid identity papers. Few pages of notebook, reveals the fate of her post separation.

"This novel is a novel. But it is also a chronicle and eye witness report." asserts the writer. Most of the characters and places are real and their 'real names' are used confirms the preface. While the theme is the genocide, he turns the mirror to the world community at large. Rwandan history until the second half of 20th century marred by Belgian rule , who spread the seed of hatred among the Tutsis and Hutus. They continue to supply arms to the warring factions ( Machettes from China the granades and firearms from French continue to enter the country by dubious means). Even at the height of the sectarian conflict, the UN peace keepers, the French or the arrogant Belgian soldiers refused to intervene. To the western world it was a news in the inner pages of the newspaper. The cry for help fallen into deaf ears, until late into the conflict. Though the peace was restored after a long time, the scars remain intact, albeit subdued and hidden.

Immediately after reading the book I watched the movie 'Hotel Rwanda' again the same night. Incidentally both claims to be based on real characters. Events in both, the book and the movie are happening at Hotel des Mille - Collins. Both happening at the same time and through two different perspective. While the movie, restricts its focus on the 1994 genocide ( a week to 10 days probably), the book has larger canvas, including the AIDS epidemic and the initial build up of the violence. Movie appealed in the visual sense, but the book seems to have been much more deep into the conflict, through the personal experience.

Very disturbing and haunting tale of violence with some graphic description of the rape, violence and death. It is difficult not to get carried away in such a tale, and he maintain a commendable restrain in his narrative. Anger, frustration, helplessness and cynicism influences the writing, despite his attempt to be im-passionate. The thin line between fiction and non-fiction often smudge. A very important recount of one of the dirtiest events of human history and not necessarily a literary phenomenon.

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A Sunday at the Pool in Kigali (2000)

Gil Courtemanche ( translated from French by Patricia Claxton 2003)

Canongate

258 Pages
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Guardian Review, Montreal Mirror, Mostly Fiction, Good Reports