Wednesday, September 24, 2014

Fermat's Last Theorem - I

Hang on. Don't give up so soon. Take heart from this feisty droplet. In a devastating book review, Peter Medawar wrote:
Just as compulsory primary education created a market catered for by cheap dailies and weeklies, so the spread of secondary and latterly tertiary education has created a large population of people, often with well-developed literary and scholarly tastes, who have been educated far beyond their capacity to undertake analytical thought.
I have been secretly fearing such a withering comment given my  penchant for indulging in sesquipedalianism. Given the title of this post, you have another reason for sending me such colourful remarks especially as there are some equations to follow. That gives me an excuse for a digression. It is just to tell you that God resides in equations so you better know something about them. As Richard Dawkins said in Unweaving the Rainbow, "What is this life if, full of stress, we have no freedom to digress.."

Apparently, once at the court of Catherine the Great, Euler met a French philosopher named Denis Diderot. Diderot was a convinced atheist, and was trying to convince the Russians into atheism also. Catherine was very annoyed by this and she asked for Euler's help. Euler thought about it and when he began a theological discussion with Diderot, he said: " (a+ bn)/n = x,   therefore God exists. Comment." Diderot was said to know almost nothing about algebra, and therefore returned to Paris.

End of digression. I would not have known much about Fermat's Last Theorem if Simon Singh had not been sued by the British Chiropractic Association. (Here is a talk by Simon Singh  where he gives some background about the problem. If for nothing else, watch the video for his hair style. Isn't it cool?) While checking out who Simon Singh is, I found out that he had written a book about Fermat's Last Theorem which I thought I will read. But as so often happens, it was a few years before I finally read it.

You must be wondering why I wanted to read about this of all things. The mathematician E.C. Titchmarsh once said, 'It can be of no practical use to know that pi is irrational, but if we can know, it surely would be intolerable not to know.'It is just a matter of curiosity and challenge. It is like the guy who was asked why he wanted to climb Everest. He replied, "Because it is there." The book is written for a lay audience so I thought maybe I can follow it.

Enough of my ramblings and on to Fermat's Last Theorem which is what I know you have been waiting so patiently for.  But first I have to tell you a fundamental idea about equations which I heard in a talk by Lawrence Krauss. I urge you all to remember it come what may:

L.H.S. = R.H.S.

Now that we are all clear about this idea, we will move on to an equation which you may have heard about somewhere: Pythagoras' Theorem. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle. This equation has an infinite number of solutions eg  3² + 4² = 5² or 5² + 12² = 13²

Fermat's Last Theorem states that in an equation of similar form, if the power is more than 2, the equation will have no whole number solutions i.e. If n is a whole number which is higher than 2 (like 3, 4, 5, 6.....), then the equation  xn + yn =  zn     has no whole number solutions when x, y and z are natural numbers (positive whole numbers (integers) or 'counting numbers' such as 1, 2, 3....). This means that there are no natural numbers x, y and z for which this equation is true i.e., the values on both sides can never be the same. What caught mathematicians' attention was that in the margin of the book where he wrote the theorem, Fermat wrote
I have discovered a truly remarkable proof which this margin is too small to contain.
That kept mathematicians busy for 300 years after his death in 1665. The brightest minds around the world struggled over the problem and some breakthroughs were achieved but a complete solution remained elusive. (As Simon Singh says in this TED talk, it even appeared in The Simpsons.)  Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. (Of course there is always the possibility that some Hindu zealot might claim that it had already been proved in the Vedic period.)

Tuesday, September 16, 2014

Startle reflex

Startle reflex is a defensive response to sudden or threatening stimuli, Sometimes when I am concentrating on a book or deep in thought about something (one of the iconic scientific images is Darwin's words 'I think' scribbled beside a tree-diagram;  my thoughts won't be so deep), if someone suddenly calls out to me, I will give a start as if a bomb has just gone off beside me. The sound won't be close to these sounds in intensity. (According to this interesting Radiolab podcast, hearing is our fastest sense.) In Laughing Gas by P.G. Wodehouse, the protagonist says:
I remember once, when a kid - from what motive I cannot recall, but no doubt in a spirit of clean fun - hiding in a sort of alcove on the main staircase at Biddleford Castle and saying 'Boo!'to a butler who was coming up with a tray containing a decanter , a syphon, and glasses. Biddleford is popularly supposed to be haunted by a Wailing lady,and the first time the butler touched the ground was when he came up against a tiger-skin rug in the hall two flights down. 
My reaction to an unexpected sound would also be as undignified as that of the unfortunate butler. My heart would jump up higher than Wordsworth's did when he saw a rainbow. It  will almost 'killofy my heart' as Epifina the bad-tempered great-grand mother of Salman Rushdie's The Moor's Last Sigh would have said. Sujit used do this often when he was smaller but he would then overdo it so the element of surprise is lost and my reactions will become normal.

A similar thing happens when slightly cold water falls on me. Sponging or any cleaning on my body is always done with lukewarm water - water that is in the Goldilocks zone: neither too hot nor too cold. I have got used to this so when normal tap water falls on me, I give a start as if ice-cold water has fallen on me.

Tuesday, September 9, 2014


In some situations, small changes stop having small effects and result in sudden qualitative changes called phase transitions. For eg., the temperature of a solid keeps increasing as you keep heating it but after a point, if you supply it with a little more heat, the crystalline structure of the solid collapses and the molecules start slipping and flowing around each other i.e. it starts melting.

Phase transitions need not occur only in chemistry. They can occur in social systems too like spreading of fads and fashions, speculative bubbles, stock market crashes, etc. The occurrence of my stroke can also be described as a phase transition. One moment I was like millions of others preparing to go  to office and from a moment later, I was unable to scratch my nose on my own. Over time, I have developed a healthy respect for itching like Ogden Nash who said:
I’m greatly attached to Barbara Fritchy; 
I bet she scratched when she was itchy.
When my nose itches, I twitch my nose and surrounding areas (part of it is involuntary) which is the signal to Jaya about what the problem is. Over time this has  become the signal for any kind of itching in any place. Through trial and error, Jaya will find out the exact spot. I am usually given head bath about once a week. By the end of that period, my head will start itching which is signal for my next head bath. At this time if my head is scratched, it feels divine. There is actually a word for the part of the body where one cannot reach to scratch.

It is not surprising that strange itching problems catch my eye. There is an article by Atul Gawande where he writes about a phantom itch:                                                                                                            
“Scratching is one of the sweetest gratifications of nature, and as ready at hand as any,” Montaigne wrote. “But repentance follows too annoyingly close at its heels.” For M., certainly, it did: the itching was so torturous, and the area so numb, that her scratching began to go through the skin. At a later office visit, her doctor found a silver-dollar-size patch of scalp where skin had been replaced by scab. M. tried bandaging her head, wearing caps to bed. But her fingernails would always find a way to her flesh, especially while she slept.
One morning, after she was awakened by her bedside alarm, she sat up and, she recalled, “this fluid came down my face, this greenish liquid.” She pressed a square of gauze to her head and went to see her doctor again. M. showed the doctor the fluid on the dressing. The doctor looked closely at the wound. She shined a light on it and in M.’s eyes. Then she walked out of the room and called an ambulance. Only in the Emergency Department at Massachusetts General Hospital, after the doctors started swarming, and one told her she needed surgery now, did M. learn what had happened. She had scratched through her skull during the night—and all the way into her brain.
I didn't know you could scratch past your skull  into your brain! Then there is an itch which  occurs when you run.  And what about Morgellons syndrome?