Thursday, September 20, 2007

the hardest math problem, ever

Fermat's Last Theorem
by: Amir D. Aczel
bookmark: resubscribe notice from National Geographic Adventure Magazine

Remember the Pythagorean Theorem? x^2 + y^2 = z^2 -- that one? It says that a square number can be broken down ino the sum of two other square numbers. Three hundred and sixty years ago, a French jurist named Pierre de Fermat, who dabbled in mathematics in his spare time (those crazy jurists!!), was reading about a related problem on breaking a square number into the sum of two squares, when he jotted this note in the margin of the very rare book Arithmetica, written by a third-century Greek named Diophantus:

On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain.


Two conclusions arise from this: First, Fermat's local library probably hated him. Second, he seems to have kind of a wry sense of humor.

More importantly, this simple statement came to be known as Fermat's Last Theorem, and went on to become perhaps the most vexing problem in all of mathematics. We don't call it his Last Theorem because it was the last one he wrote--we're not even sure when he wrote it, and that copy of Arithmetica has long since been lost to the ages. It's referred to as his last theorem because all of his other theorems had already been either proven or disproven--decades or even centuries ago. This one stumped everybody.

In fact, it proved so vexing that it would take over three hundred years and the work of dozens of mathematicians working all fields of mathematics to solve it. So vexing that entire new branches of mathematics, even new numbers had to be invented before we could even approach the solution.

Aczel manages an incredible task. He collapses four millenia of mathematical history into 136 pages (the book is hardcover, and I carried it around in my back pocket for a couple days) that are... gripping. I'll readily and happily admit that I'm probably the biggest geek on this blog (we have lots of book geeks, a law geek, a theater/singing/dance geek, and a champion of libraries, but I'm a straight-up math and science geek. ~e is my closest competition here, and I think even she'd agree I have an edge), but I'm also fond of telling people that I'm not very good at math. You don't have to be. Anybody who's made it through high school algebra, and maybe a touch of calculus, can understand this book. Aczel explains everything you need to know to understand how Andrew Wiles eventually developed a solution to the Hardest Math Problem Ever. When he gets to the more advanced stuff--math that is truly understood by maybe a dozen people in the entire world--he gives an explanation that is general enough to give the reader a basic idea of what's happening, without actually going into so much detail as to lose anyone. For all the math explained in this tiny book, there's hardly any numbers anywhere. Aczel could explain these concepts to an English major, and they'd still come out of it feeling more enlightened than befuddled.

That's what is most fascinating to me about this book. I mean, sure--the book has it all. Political intrigue, suicide, deception, theft, deceit, betrayal, centuries of history and scientific progress, secret societies, and lots and lots of math, but it's crammed neatly and tidily into a very small volume. It's easily readable by anyone with a good grasp of math, and not too daunting for anyone else. The book is entirely unlike the problem, which had everybody stumped for so long that many of the brightest mathematical minds this world has ever seen simply dismissed it, believing that it would either never be solved or at least not in their lifetimes.

Aczel writes:

What is interesting about Fermat's Last Theorem, however, is that it spans mathematical history from the dawn of civilization to our own time. And the theorem's ultimate solution also spans the breadth of mathematics, involving fields other than number theory: algebra, analysis, geometry, and topology--virtually all of mathematics.


And in that, the book is exactly like the problem

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posted by reyn at 7:58 AM

7 Comments:

Blogger Kat said...

You write "English major" with disdain, like English majors are none-too-bright. I take offense at that. :P

9/20/2007 9:01 AM  
Blogger reyn said...

We each have our specialties. English majors are not known for their mathematical prowess; engineers are rarely lauded for their poetry. I see you didn't even notice that I touted you as a "champion of libraries," and I was rather proud of that line.

9/20/2007 9:08 AM  
Blogger Kat said...

Yes, champion of libraries, I did quite like that. I meant to comment on it - it's a great description. I'm also quite happy being referred to as as library nerd too. :)

And for an English major, I did quite well in math - and I enjoy it too. Of course, the only math I've taken since high school was stats. We did all the statistical analyses by hand, and I liked it! :P

However, I suppose in general English majors are not known for their math schools.

9/20/2007 9:16 AM  
Blogger reyn said...

To every rule, there are exceptions.

I hate statistics with a fiery, all-consuming passion. You can have mine. I'll trade you for your sandwich.

9/20/2007 10:16 AM  
Blogger Elizabeth said...

As an English major, I must say I'm rather hurt by your evident disdain, reyn! (Although I can take comfort in that Fermat the mathematical genius was a compatriot jurist.) I'm good at math! I took Calc 2 in college, and can do all the computing I need to simply by counting on my eleven fingers.

Question: Do we think the solution we found is the same as Fermat's? It seems like his must've been simple enough that "new numbers" didn't have to be invented. In other words, are people still looking for a more elegant solution?

[And speaking of statistics, Kate...do you recall that oh-so-memorable afternoon of statistics grinding at your house during high school? Me, you, and G. I don't remember at all what our project actually was, except that in our final summary, I wrote that I'd prefer to get a degree in underwater basket weaving to ever studying statistics again!]

9/20/2007 9:50 PM  
Blogger reyn said...

We know that our solution is not Fermat's, because the therorems, numbers, and (since proven) conjectures Wiles used to solve FLT simply didn't exist in Fermat's time. There are three ways to consider this. First, hes was wrong, and didn't have a proof. Second, he had an idea for a proof, and hadn't fully tested it yet. Third (this is most likely), he was referring to his own method of Infinite Descent (http://en.wikipedia.org/wiki/Infinite_descent), which works for the case of n=3 (where n is the exponent), but not on anything else. He may have discovered the n=3 case, and believed that it was a general solution, and never got around to proving himself wrong.

Since we don't have many of his notes (like his copy of Arithmetica), there's simply no way of knowing what he might have worked out.

9/21/2007 6:18 AM  
Blogger reyn said...

Wait--you two were grinding each other?? Do you have any idea how much some people would pay to see that??

9/21/2007 6:19 AM  

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