Mathematics Thread for David

[halfwise]

I've decided David needs his own thread to expound upon the beauties of mathematics (and hence distract him from horse colors, which is clearly a sad waste for a brain the size of a planet). So I've made one just for him. I'll kick it off with what just popped up in the news:

It seems the ABC conjecture (which I had never heard of before but appears to be important) may have just been proved so that it can be elevated to a theorem. The lay article I read states that Fermat's last theorem and a host of other number theory stuff follows directly from it.

Why is this somewhat odd theorem so important?

[Mrs Figg]

funny you should ask.. ha ha just kidding...no idea wot you said...

[halfwise]

Me either, that's why I'm fishing for David.

[David H]

Why is Fermat's last theorem so important? Well for that question to have meaning you must first assume that mathematics itself is important. From there, everything else is a corollary.

Where to start....? The history of diophantine equations and the first solutions of the Pythagorean theorem?

OK everybody knows that x^2 + y^2 = z^2 has an infinite number if integral solutions, right? In other words there exist an infinite number of right triangles whose side are perfect integers. These Pythagorean triples were fundamental to ancient Egyptian surveying, and are at the foundation of all our architecture, mapping, charting, and measuring the universe.


So it seemed natural to mathematicians throughout the centuries that really cool things would happen if you could prove something equivalent for x^n + y^n = z^n where n>2. Except it doesn't work. There just aren't any integer solutions for higher powers, no matter how hard you look! Then the question became "why not?"

In 1636, Fermat wrote in the margin of one of his books that he'd found a marvelous proof of "why not" but didn't have room to write it at the moment. Then he died. This triggered the largest wild goose chase in the history of mathematics. Gauss pretty much revolutionized modern algebra in an attempt to crack this nut, failing miserably and publicly, but defining much 19th and 20th century mathematics in the process.

The proof finally fell to Andrew Wiles in 1995. There's a good NOVA episode on it. I tried looking at his proof when it came out, but it's way over the head of a humble farmer. It's technically proven, but it's far from elegant.

The ABC conjecture, which I've just been looking at, seems to provide a much more elegant, intuitive proof which should suggest even more wild geese for mathematicians to chase in the future.

But is it important? I guess it depends on how much you like wild geese....

[Mrs Figg]

with taters?

[Pettytyrant101]

Amazingly David I think after reading your explanation I am now more ignorant than I was when I didnt know anything.
Is that what mathmaticans feel like all the time?

[halfwise]

A more intuitive proof of Fermat's last theorem would be nice, but if it's based on another 500 page proof I wonder how much is actually gained.

[Pettytyrant101]

Not to turn this thread to Who but the Doctor claims it his fault the theorem was not completed as he overslept the day Fermat died.

[David H]

Amazingly David I think after reading your explanation I am now more ignorant than I was when I didnt know anything.
Is that what mathmaticans feel like all the time?

My uncle once explained education to me this way:

When you think you know everything, they give you a high school diploma.
When you realize that you don't know everything, they give you a Bachelor's degree.
When you realize that you don't know anything, they give you a Master's degree.
When you realize that you don't know anything, but neither does anybody else, they give you a Doctorate and ask you to teach people.

[David H]

A more intuitive proof of Fermat's last theorem would be nice, but if it's based on another 500 page proof I wonder how much is actually gained.

The gain really has nothing to do with proving Fermat's theorem yet again. Nobody has had any worries about the actual truth of the conjecture. It's all about sharpening the tools that were forged for the proof, and then seeing what else they may imply. That's how you peel the onion, one layer at a time.

[halfwise]

I guess if the ABC theorem was somewhat intuitive but hard to prove, it would be worth using for other things. People could see their way to other proofs more easily.

[David H]

I looked a little deeper. This proof uses some of the same approaches as Wiles' proof of Fermat but apparently solves the problem in a much longer and more general way. Very exciting because it's at that state where nobody can guess it's implications. I don't know enough to have an opinion, bit it seems to contain a fundamental new insight into the nature of prime numbers. That could certainly have implications for quantum physicists, if that's what you're asking.

{{once again I fear that mathematicians are sweating blood over a period of years to forge precision cutting tools, while the physicists sit back saying, "I bet I could really cut stuff with that if I hit it with a big enough hammer."}}

[Mrs Figg]

its all becoming clear. kinda.