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.)

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