Margins of a theorem4 min read . Updated: 19 Aug 2011, 02:04 PM IST
Margins of a theorem
Margins of a theorem
On 17 August 410 years ago, a man was born in France whose name tormented mathematicians for most of those 410 years. On 17 August this year, another of Google’s famous home page doodles celebrated him. Yet it wasn’t, as you might expect, a portrait of the man. It was instead a few mathematical-looking symbols, and a reference to the most famous margin note in mathematical history.
You see, mathematicians don’t need to know what the man looked like. Pierre de Fermat’s Last Theorem (FLT) will do just fine.
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Edmund Landau, a German mathematician, used to stock copies of a brief form letter: “Dear Sir/Madam: Your proof of Fermat’s Last Theorem has been received. The first mistake is on page _____, line _____." Useful, because crank amateur “proofs" of FLT once constituted a near-cottage industry. No more copies today: not just because Landau died in 1938, but because FLT was actually proved in 1995, by an Englishman called Andrew Wiles. Though don’t look for the proof here: this space in Mint is too small to contain it, not that I would understand the proof anyway.
Yet FLT’s appeal, the reason amateurs found it so attractive, is that it is so easily understood. You need no mathematical training. You need only to 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) = 2x2x2 = 8. Or 7 (7 to the power of 4) = 7x7x7x7 = 2,401.
Now consider this:
x² + y² = z²
(This will be familiar if you remember Shri Pythagoras of ancient Greece and your school texts). There are many trios of integers that you can use for x, y, and z here. For example, 3² + 4² = 9 + 16 =25 = 5². Or 5² + 12² = 25 + 144 = 169 = 13²
With me so far? But what if you changed that power to 3 or more? Can you still find integers to substitute for x, y and z? Ah, now you’re squarely in FLT land: get ready for a surprise.
If the power is any number greater than 2, you can never find integers for x, y and z. That is, more formally, xn + yn = zn has no integer solutions when n is greater than 2.
Still with me? Well, when Fermat read about this in a textbook in 1637, he made that most famous margin note in mathematics: “I have discovered a truly remarkable proof which this margin is too small to contain."
Unfortunately, Fermat never published his “remarkable proof". His scribbled words have tormented mathematicians, amateur and otherwise, ever since. For 350 years and more, nobody could prove what came to be called Fermat’s Last Theorem. It was such an intractable problem that mathematicians began to suspect Fermat had not actually found a proof. Perhaps, he tried his hand at it for a while, thought he was close enough to a proof that it didn’t matter, and went on to other things. Perhaps, it was just an elaborate Fermat-ian joke.
The irony was that Fermat himself was really a lawyer, and only an amateur mathematician. Nearly every eminent mathematician—maybe some lawyers too—of the last four centuries has tried to tackle FLT. They have proved various related theorems, each seeming like a step towards the FLT holy grail. But oddly, of these intermediate steps, Andrew Wiles used just one.
In 1955, Yutaka Taniyama examined the properties of certain mathematical objects called elliptic curves. Among other things, these curves are useful in the arcane art of code-making: like, for example, encrypting credit card numbers when you buy online. Taniyama and two colleagues suggested a specific property of these curves, called the Shimura-Taniyama-Weil Conjecture (STW). In 1986, STW was shown to be connected to FLT: if STW was proved for a particular set of curves, FLT would also be proved.
Which showed that FLT was not some trivial curiosity, but has implications for, among other things, cryptography.
It was Wiles who finished the job. In 1993, he proved STW for those particular curves, and so solved mathematics’ greatest puzzle. There were some major gaps in his work, but by 1995, he had filled those too. Like Fermat, Wiles needed more space for his proof than a margin could contain. It sprawled across all 130 pages of the May 1995 issue of the journal Annals of Mathematics.
No, there was no form letter in the mail from Edmund Landau.
Wiles presented his results over three lectures at Cambridge University in June 1993. When done, he wrote Fermat’s Last Theorem on the board, said “I will stop here," and sat down.
What I would not have given to have been there right then. Google’s doodle? Nice, but it’s a poor substitute.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at email@example.com