May 22, 2005

Marginal at best

For many years, Edmund Landau, a German mathematician, had a form letter that looked like this: "Dear Sir/Madam: Your proof of Fermat's Last Theorem has been received. The first mistake is on page _____, line _____." Landau would assign the job of filling in those blanks to one of his students. They must have been busy, because crank "proofs" of Fermat were something of a cottage industry; if I recall right, Orissa was a minor breeding ground for them.

But Landau's form letters are likely gathering dust today. Not just because the man passed on a long time ago, but because Fermat's Last Theorem (FLT) has actually been proved. Undoubtedly the best-known unsolved problem in mathematics for over 350 years, this little conundrum was finally solved in 1993 by an English mathematician called Andrew Wiles. This marginal essay is too small to contain the mathematics of Wiles' proof, and I wouldn't understand it anyway. But many crank talents now like to think they do: the cottage industry has turned to claims of disproving Wiles. Time to resurrect and change those form letters, I think.

FLT's appeal, the reason it was such a target for amateur mathematicians, is that the problem is so easily understood. You don't need any mathematical training to follow it. You need only know about raising a number to a power -- multiplying it by itself a certain number of times. For example, 2³ (2 raised to the power of 3) = 2 x 2 x 2 = 8. Or, 54 (5 raised to the power of 4) = 5 x 5 x 5 x 5 = 625.

Take this equation:

x² + y² = z²

(This should be familiar to those who remember a certain Shri Pythagoras and the properties of right-angled triangles). There are many trios of numbers that you can use as x, y, and z here. For example, 3² + 4² = 9 + 16 = 25 = 5², or 5² + 12² = 25 + 144 = 169 = 13² .

But what if you changed that power n to 3, or more? Now you're in FLT territory, and you bump into something startling. Take this equation:

xn + yn = zn

FLT says that if n is greater than 2, there are no numbers that will fit x, y and z. In other words, you cannot find three numbers such that if you add the nth powers of two of them, you will get the nth power of the third -- if n is greater than 2.

Reading about this theorem in 1637, Pierre de Fermat made the most famous margin note in mathematics: "I have discovered a truly remarkable proof which this margin is too small to contain."

But Fermat never published his proof. The few scribbled words have tormented mathematicians, eminent and otherwise, ever since. Nobody could prove the theorem. It was such an intractable problem that mathematicians began to suspect Fermat had not actually found a proof. Perhaps he simply tried his hand at it for a while, found that standard methods were not working, and went on to other things. Perhaps he thought he was close enough to a proof that it didn't matter. Perhaps it was all an elaborate Fermat-ian joke.

Mathematicians who have tried to tackle FLT -- nearly every eminent one of the last three centuries -- have proved various related theorems. Each seemed like a step towards the Holy Grail. But with one exception, Wiles' solution was curious in that it didn't use any of these intermediate steps.

The exception was the work of two 19th Century German mathematicians, Kummer and Dirichlet, a cornerstone of modern algebra and the foundation of Wiles' proof. But the final step in the saga actually had its origins in the work of Yutaka Taniyama. In 1955, he was examining the properties of elliptic curves. With two others, he suggested a specific property of these curves, what's called the Shimura-Taniyama-Weil Conjecture (STW). In 1986, STW was shown to have connections with FLT: if STW was proved for a particular set of curves, FLT would also be proved.

(This connection showed that FLT is not some trivial curiosity of number theory, but has a lot to do with fundamental properties of curves and space).

It was left to Wiles to bridge the gap. In 1993, he proved STW for those particular curves, and so solved mathematics' greatest puzzle. There were some gaps in his work, but over the next couple of years, he took care of those. His eventual proof, like Fermat's, was also a little longer than a margin could contain: it actually filled an entire 130-page issue of the journal Annals of Mathematics.

And it's likely he did not dispatch his proof to Edmund Landau.

Wiles presented his results in a series of three lectures at Cambridge University in June 1993. At the end of the final lecture, he wrote Fermat's Last Theorem on the board, said "I will stop here," and sat down.

Boy, I'd have loved to have been there at that precise moment.


Kartik said...

Funny you should mention it ... I read a most intriguing book on this about a year back, it was called "Fermat's Last Theorem - The most intriguing problem in mathematics" or something to that effect ( I know for certain it wasn't that exactly ). They go on to discuss it right from the time of hte Greeks down to Andrew Wiles.


Anonymous said...

Marginal at best! This is perhaps the best title that I've seen recently.

Nice post, Dilip.

Anurag said...

I'd have loved to be there, too. Sigh.

You seem to like science and mathematics a lot, Dilip. Have you read "Schrodinger's kittens"? It's a must read...

Anonymous said...

Dear Dilip,

Perhaps you could have emphasised that
the you are talking only about "natural
numbers" when trying to find solutions to
your equations. Otherwise,

(root 2) squared plus (root 2) squared= 4

is always possible. Incidentally, the best popular account of the Fermat Saga is due to Simon Singh (Fermat's enigma). Also due to him is an excellent BBC documentary on the subject.

Dilip D'Souza said...

Ravi, I almost picked up Simon Singh's book some months ago, after reading a good review somewhere. Don't know why I didn't. But now that you've mentioned it again, I will make sure to get it. (Kraktik, is that the book you had in mind?) Have you read A History of Pi"? Quite a delight.

Also, you're right about natural numbers, but the thing about these articles is, I need to limit what I will explain, what concepts I write about. I took the easy way, assuming that most people who read this would understand "number" to mean "natural number".

One more thing: while I get your point, the solution you offer is not what the equation (Fermat's) asks for. The theorem holds for powers greater than 2.

Anurag, I have not read "Schrodingers Kittens", tell me more. Why not make a trip from Pune here and drop it off...?

Fadereu old chum, "like a good teacher"?! I think I had better revise how I write about science after that! I know little about Euler's Identity, despite your Wiki pointer. Anand, you wanna be the teacher?

Suresh Venkatasubramanian said...

Hi Dilip,
you missed the coda. As it turns out, Wiles was not done. there were serious problems in his proof that needed to be fixed, and were finally resolved a few years later in papers by Wiles and Taylor.

More here

Nice job on the article. Your science writing is infrequent but always a pleasure. as a math blogger, I appreciate the company :)

Anonymous said...

Dear Dilip,

A follow up on my earlier comment about natural numbers. First

(cube root 4)cubed plus (cube root 4)cubed
=2 cubed

would always give you a solution for n=3. And you can do this for any power. I picked squares just to show that even finding the Pythagorean triplets reduces to a trviality if one allows arbitrary numbers on the real line as possible solutions. It is restricting the problem to natural numbers that makes it really hard.

To give a more precise history: Wiles announced the proof of his theorem in 1993. There was a serious gap. It was resolved in August 1994 by a the paper of
Rcihard Taylor and Andrew Wiles in the same volume of the Annals of Mathematics cited by you. Incidentally, C. Khare (at that time in TIFR, Mumbai) and R. Ramakrishna subsequently developed their own approach to the Taylor-Wiles systems,
as they are known. C. Khare's work has recently gone "beyond" FLT. Check out


Anonymous said...

Rohit -- There may not be more to talk about that identity than what Wiki does.

Suresh -- I thought Dilip did not miss that point. In fact he stressed that point:

There were some gaps in his work, but over the next couple of years, he took care of those. His eventual proof, like Fermat's, was also a little longer than a margin could contain: it actually filled an entire 130-page issue of the journal Annals of Mathematics.

Also he linked to Wiles and Taylor-Wiles.

Larry Freeman said...

Really enjoyed this blog!

I recently started a blog that you might be interested in (it is not as well written).

I am an amateur who is using a series of blogs to trace the history of Fermat's Last Theorem from Fermat's notes in the margin to Wiles' proof. My focus is on the mathematics and proofs. The goal is to present a series of proofs in the same format as Euclid's Elements.

I would be very interested in any comments that you have:



Anonymous said...


Many Many thanks for discussing mathematics and science in general. I find very lonely in my land where none is interested in talking science.
You have done very good background reading, I'm happy, otherwise you would not have known the famed Taniyama-Shimura conjecture.
It is also worth mentioning that by the time he got a conclusive proof, Wiles crossed that magic age-limit of 40 for Fields medal, but still got one special prize.

If it interests you, there is another "star work" in the horizon sone by a young Russian Mat'cian Grisha Perelman. This is also a proof of a famous conjecture - Poincare' conjecture in geometry. I'm more familiar with this than Fermat's theorem. Still if you could write a few words about this in future, will be grateful.

A student of theor. physics.

Anonymous said...

Here's more information on a new proof of Fermat's Last Theorem, due to Khare, Wintenberger and Dieulefait (it also mentions a related conjecture of Serre):