Sunday, October 26, 2014

Some interesting properties of numbers

In  Fermat's Last Theorem, Simon Singh writes about some interesting patterns among numbers:

  1. Perfect numbers - Numbers whose divisors add up to the number itself. Eg.6 has divisors 1, 2 and 3 which add up to 6. The next perfect number is 28. If the sum of the divisors is more than the number, Pythagoras called it an 'excessive' number and if the divisors added up to less than the number, he called it 'defecttive'.
  2. Friendly numbers or amicable numbers are closely related to perfect numbers.They are pairs of numbers such that each number is the sum of the divisors of the other number. For eg. 220 and 284 are friendly numbers. 220 is the sum of the  divisors of 284 (1, 2 4 71 and 142). 284 is the sum of the divisors of 220 (1, 2, 4, 5,10, 11, 20, 22, 44, 55and 110). Fermat discovered 17,296 and 18,416. Descartes discovered a third pair (9,363,584 and 9,437,056). Leonhard Euler discovered 62 pairs. Strangely,they all had missed a much smaller pair which was discovered by a 16 year old Italian - 1184 and 1210.
  3. Sociable numbers are 3 or more numbers which  form a closed loop. Consider the loop of five numbers: 12,496; 14,288; 15,472; 14,536; 14,264. The divisors of the first number add up to the second, the divisors of the second add up to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.
  4. Fermat proved that 26 is the only number sandwiched between a square and a cube (between 25=52  and 27= 33)
  5. All prime numbers (except 2) can be placed in two categories: those  which can be written as 4n + 1 and those which can be written as 4n - 1 where n equals some number.Thus 13 is in the former group (4*3 + 1) while 19 is in the latter  group (4*5 - 1). Fermat's prime theorem claimed that the first type of primes were always the sum of two squares while the second type could never be written in this way. The theorem was proved by Euler almost a century after Fermat's death.

PS: Since I will rarely do any maths related posts (I know that you are dreadfully disappointed but as a wise philosopher of yore said, you can't always get what you want) I am generally giving a couple of links about maths that I had saved:

  1. An excellant series in NYT by Steven Strogatz on The Elements of Math
  2. A talk by Simon Singh about "The Simpsons and Their Mathematical Secrets

Wednesday, October 15, 2014

Fermat's last theorem - III

You thought you had escaped a bad dream with the last post, didn't you? But the title of this  post would have revealed to you that, like George Bush, you were a bit hasty in arriving at the conclusion that the mission had been accomplished.

Where we last left him, Andrew Wiles had become a celebrity but the peer review process was only beginning. This is a process that all scientific works have to go through before being accepted as correct. Wiles had to submit a complete manuscript to a leading journal.whose editor will then choose a team of experts who will then examine the proof line by line. The proof was so complicated that 6 referees were appointed with each given responsibility for one section of the proof. The process took a few months.

At first all the problems were minor and Wiles sorted them out quickly but there was one 'little problem'. And you are right - it had  to do with the Kolyvagin-Flach method which he had used in the proof. No matter how hard he tried he couldn't fix the problem. If he fixed the problem in one place,another problem cropped up somewhere else. The issue dragged on for months. (But tell me how did you guess it was the Kolyvagin-Flach method?)

Rumours started flying in the mathematical community - perhaps this was another in the long line of failed proofs for Fermat's Last Theorem. Wiles wanted to work in isolation and concentrate completely on the problem but this was not possible since he had become a celebrity. There was pressure on him to publish the incomplete proof so that someone else could try to correct the flaw but this would have meant the end of a childhood dream.

In desperation he took on a collaborator and struggled on for a while but the problem seemed intractable. He was on the verge of giving up when  one day in Sept. 1994, over an year after his initial presentation, he had an inspiration - all he had to do  to make the Kolyvagin-Flach method work was to use it in conjunction with the Iwasawa theory! Aren't you amazed by the man's brilliance? Simon Singh gives the views of a mathematician on the final 130 page proof in Fermat's  Last Theorem :
'I think that if you were lost on  a desert island and you had only this manuscript then you would have a lot of food for thought. You would see all the current ideas of number theory. You turn to a page and there is a brief appearance of some fundamental theorem by Deligne and then you turn to another page and in some incidental way there is a theorem by Helleguarch - all of these things are just called into play and used for a moment before going on to the next idea.'
Oh....Ah.. That Sounds Very Interesting. (As Douglas Adams said in Dirk Gently's Holistic Detective Agency , "Capital letters were always the best way of dealing with things you didn't have a good answer to.") But... er, can I exchange it for a Wodehouse? Much obliged.

But the question is, was Wiles' solution the same as Fermat's? It couldn't have been because Wiles' proof involved 20th century mathematics which had been unavailable to Fermat so Fermat's proof, if it existed, must have been simpler.

Believe it or not, I am through with Fermat's Last Theorem. Now you can safely breathe a sigh of relief, grab a restorative  drink and check Facebook.

PS: As a reward for your patience, here is a bit of nonsense math.

Sunday, October 5, 2014

Fermat's last theorem - II

Computers had been used to check Fermat's Last  Theorem (Fermat's Last Theorem states that xn + yn =  zn    has no whole number solutions for x, y and z when n > 2) for the first trillion numbers and it had proved to be true. For most of us this would have been enough but mathematicians are a finicky lot. There still remained infinite numbers to be checked so the brute force of a computer could never be used to prove the theorem. Simon Singh relates a story in Fermat's Last Theorem to illustrate mathematicians'penchant for exactness:
An astronomer, a physicist and a mathematician (it is said) were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field. 'How interesting,' observed the astronomer, 'all Scottish sheep are black!' To which the physicist responded, 'No, no! Some Scottish sheep are black!' The mathematician gazed heavenward in supplication, and then intoned, 'In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.'
Enter Andrew Wiles. As a 10 year old he saw the theorem in a library book and was fascinated by it. (You would have thought that 10 year olds had other things to do but there it is.) That fascination became an obsession for the next 30 years before he finally cracked it. For the last 7 years, he shut himself off from the rest of the world to focus solely on the problem. Nobody in the world knew what he was doing. This was highly unusual in the field  of mathematics where people frequently checked new ideas with each other.

Wiles' proof involved deadly beasts like elliptic equations, modular forms, L-functions, Taniyama-Shimura conjecture etc. I know you want to know more about these things but since I know only their names and nothing else, I must respectfully avoid throwing any light on them. Browning said that one's reach should be beyond one's grasp or what is heaven for? I am sure he meant well but I have the unfortunate tendency of steering well clear of things that are way beyond my reach. What to do, I am like that only (sic). A thousand apologies!

But I know that you will not let me live in peace till I give you some idea of how he went about his quest so here is my honest effort. Elliptic equations and modular forms are two widely separated areas of mathematics that didn't seem to have any obvious connection with each other like say, probability and calculus. Then the Taniyama-Shimura conjecture was proposed out of blue stating that thees two forms were actually two different manifestations of the same underlying property. Wherever it was checked, it proved to be true but it remained a conjecture. It faced the same problem that Fermat had: how to check for infinite possibilities?

It was then shown that if Taniyama-Shimura was right then Fermat was right i.e. either both were right or both were wrong. So Wiles decided to try to prove Taniyama-Shimura. If he could do that then Fermat was right by default. During the final stages of the proof he began to wonder if he was on the right right track. He decided to confide in another mathematician, Nick Katz. They decided they would design a lecture course for graduate students which Katz would also attend and they would check the calculations. Simon Singh writes:
'So Andrew announced this lecture course called "Calculations on Elliptic Curves",' recalls Katz with with a sly smile, 'which is a completely innocuous title -it could mean anything. He didn't mention Fermat, he didn't mention Taniyama-Shimura, he just started by diving right into doing technical calculations. There was no way in the world that anyone could have guessed what it was really about. It was done in such a way that unless you knew what this was for,then the calculations would just seem incredibly technical and tedious. And when you don't know what the mathematics is for, it's impossible to follow it. It's pretty hard to follow it even when you do know what it's for. Anyway, one by one the graduate students just drifted away and after a few weeks I was the only person left in the audience.'
After some more time and a few more steps (sounds simple, doesn't it?) Wiles was ready with the proof. He decided to announce it at the Isaac Newton Institute in Cambridge during a workshop called 'L-functions and Arithmetic' where he was slated to give 3 lectures called 'Modular Forms, Elliptic Curves and Galois Representations' (there are people who go to such conferences and attend such lectures) which was later called 'The Lecture of the Century'.  There were rumours circulating that some big result was going to be presented but Wiles didn't let on.

After the first 2 lectures the audience was still not sure whether he would actually come up with a big result. Finally towards end of the 3rd lecture he read out the proof, wrote up Fermat's Last theorem and said, "I think I'll stop here." And then the audience burst into applause - he had solved a 350 year old problem. E-mails were flying and soon newspapers, TV crews and science reporters descended upon the institute wanting to interview the 'greatest mathematician of the century'. He had become a celebrity. Simon Singh writes:
This was the first time that mathematics had hit the headlines since Yoichi Miyaoka announced his so-called proof in 1988: the only difference this time was that there was twice as much coverage and nobody expressed any doubt over the calculation. Overnight Wiles became the most famous , in fact the only famous, mathematician in the world, and People magazine even listed him among 'The 25 most intriguing people of the year' along with Princess Diana and Oprah Winfrey. The ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.