Fermat's Theorem

Factual Questions
21

Well, first of all, most people who call themselves mathematicians spend most of their time teaching non-mathematicians the math necessary for their own jobs. Even leaving aside math teachers at high school level and below (who extremely rarely do any mathematical research of any kind), the people who call themselves mathematicians at institutions above that level are nearly always teaching, not doing research. If you teach at a community college, you’ll be lucky to have any time to do research, and the college knows that and doesn’t consider it to be an important part of your job. Very few of your students will become mathematicians themselves. As you move up on the scale of a university’s prestige, there will be more and more opportunity to teach students who will become mathematicians themselves and more and more opportunity to do research, but very rarely will it become a major part of your time. If you work as a mathematician in industry or government, you will be assisting other scientists and engineers in their jobs. Some of what you do will be doing is research and will often be publishable research, but the research you do on work time will not be pure mathematical theorizing. So get out of your head the notion that even those people who call themselves mathematicians spend some significant amount of time on average doing pure mathematical research with no clear use in other fields. The amount of that is pretty small compared to the total amount of time they spend on other things in their jobs.

Furthermore, there’s no way to tell for sure if a particular piece of pure mathematical research will or will not have some use outside of mathematics. Often a piece of mathematics created with no particular use in mind will be seen decades or centuries later to have an important application to another field. aruvqan, do you realize how offensive you are being? It’s O.K. if you find math difficult, but don’t assume that you’re qualified to say how useful it is for other people who do understand it better.

22

Step 1: get accepted to a graduate program in mathematics.

This completes the process.

In all seriousness I often spent several hours a day in grad school thinking about math I didn’t know how to do. Sometimes that process terminated with a new understanding and frequently it did not. Now I’m moving on to a post doc and I don’t expect any of the above to change a whole lot.

23

Here’s a weird thing related to Fermat’s theorem that I never see
It’s true that there are infintely many solutions to

**a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup] **

for **a, b,**and c integers, but no solutions for

**a[sup]3[/sup] + b[sup]3[/sup] = c[sup]3[/sup] **

, or for any higher powers in which the numbers are integral values

BUT
There ARE solutions for

**a[sup]3[/sup] + b[sup]3[/sup] + c[sup]3[/sup] = d[sup]3[/sup] **
for which a,b,c, and d are all integers, the smallest being

3[sup]3[/sup] + 4[sup]3[/sup] + 5[sup]3[/sup] = 6[sup]3[/sup]

There’s also

1[sup]3[/sup] + 6[sup]3[/sup] + 8[sup]3[/sup] = 9[sup]3[/sup]
AND

There are no sums of two fourth powers for which the rsult is the fourth power of an integer, but there are sums of four fourth powers of integers for which the result is the fourth power of an integer, starting with:
30[sup]4[/sup] + 120[sup]4[/sup] + 272[sup]4[/sup] + 315[sup]4[/sup] = 353[sup]4[/sup]
and so on. It seems as if a pattern is developing here, but it breaks down by the eighth power or so, which I think has a counter-example with a sum of only seven other eighth powers.

24

And to be honest, I was trying to be silly.

Sheesh, unstuff your shirt :frowning:

25

Continuing the hijack:

This is called the Jordan curve theorem, for anyone interested in looking up further details. I don’t know of a simple direct proof, but it follows immediately from Alexander duality. It requires a decent bit of background to prove (Alexander duality is part of the canonical first-year grad class in algebraic topology), but it’s not an unsolvable or even particularly difficult problem with modern machinery.

Anyway, to address the original post:

Nope. The proof is a nontrivial consequence of the (for more difficult and consequential) Taniyama-Shimura conjecture, which states that every rational elliptic curve is modular. Expanding that last clause is tricky, and I don’t think I can do a better job than Wikipedia.

26

Though, as MikeS noted, the Jordan curve theorem (which is always trotted out as an example for this sort of thing) is not particularly difficult to prove from appropriate first principles for suitably nice curves; e.g., polygons, or even piecewise smooth curves more generally.

It only becomes difficult to establish for technical definitions of “curve” that encompass a spectrum of pathological monsters, about which the theorem might well be said to have never been obvious anyway. Even then, the proof can be made conceptually straightforward and elementary, if messy in the details (messy because so are the details of the definition of “curve” being used): establish the theorem for polygons as before, and then establish that polygons can suitably well approximate the other curves of interest as to allow the theorem to be carried over to this greater generality. [See, for example, any of the proofs of the Jordan curve theorem via so-called “nonstandard” analysis]

(Of course, the homology based proofs provide greater insight and generalization, but it is only to be expected that greater insight and generalization comes with the development of further abstract machinery; this does not, I think, refute the possibility of considering the intuitive claim reasonably elementary to establish.)

27

Indeed. (As you know, Wendell) G. H. Hardy went to his grave believing that his work had no practical applications, but today his work is used in physics, and in encryption theory.

28

Then again, one might fairly say that mathematicians like pathological monsters.

29

Jonathan Coulton would disagree - “Pathological monsters, cried the terrified mathematicians, every one of them a splinter in my eye”…

30

Here’s what Andy L is talking about:

31

And sometimes a piece of mathematics created with a particular use in mind centuries ago, taught to all students decades ago will be reinvented by someone and show how bad at maths people are in some fields:

Medical researcher discovers integration: gets 75 citations.

32

Yep, I realized a little after asking that question that probably a lot of mathematics has played out like that. :smack:
I recall that even number theory did not have a strict, formal foundation before set theory.

33

Well all this is doing is reminding me that I’ve forgotten a lot of my BSc in Pure Maths that I completed in 1996.

Thanks guys. :wink:

34

To be fair I thought you were kind of insulting as well. I can see you don’t like math, which is your right, and your opinion of mathematicians is affected by that, but we are doing legitimate work, you know. And we don’t get off on making people like you suffer in high school! :stuck_out_tongue: (High school math teachers are not mathematicians anyway.)

Anyway, no harm done. :slight_smile:

35

Yeah, I can easily understand why mathematicians might take umbrage at some of your comments, like:

I can think of at least two ways that you’re poking mathematicians with a stick here.

(1) For literally millennia, mathematicians have resented it when people demand, “But what use is it?” as though truth had no inherent value.

“Give that man a coin, since he must make a profit from what he learns!” - Euclid

“Of what use is a newborn baby?” - attributed to various famous scientists

(2) Belittling what you don’t understand. Your last line, especially, conveys the attitude that if you can’t understand something, it must be esoteric mumbo-jumbo of interest to no one who isn’t hopelessly geeky.

This is an attitude that plagues not only mathematicians but also artists, writers, musicians, etc. People try to read a complex novel or look at an abstract painting and, if it’s over their head, they conclude it must be overrated, worthless, elitist crap.

Now, as for how “a formula without a solution” could matter to anyone, consider this example of a theorem that’s similar to Fermat’s Last Theorem only much, much simpler:

The equation a + b = c has no solutions for which a, b, and c are all odd numbers.

Could knowing this fact ever come in handy? Possibly. But, more importantly, if you can prove it—which means, if you can understand why it’s true—you have a better understanding of how numbers work in general, and the approach or techniques you developed to prove it might be useful in proving more substantial things as well.

36

As evidenced by the large number of books of counterexamples in various fields of mathematics. I’ve always wanted to compile a bunch of these into a single volume and call it “The Big Book of Mathematical Pathology”.

A mathematics professor is giving a lecture, and states a theorem that he wants to prove. He starts in on his proof, but halfway through a student in the front row raises her hand. “Wait,” she says, “that proof can’t work — I can think of a counterexample to your theorem!” “That’s OK,” the mathematician replies, “I have another proof.”

37

Fermat was very clever. i rate hime one of the greatest minds in the last 500 years. proving his theorem was difficult but have you guys ever looked at my Theorem? Armstrong’s last Theorem?:dubious:

38

Rather presumptuous to name it your “last” theorem. Are you planning to die soon?

39

You shouldn’t call it the Last Theorem. All Great Laws come in sets of three – Euler’s Three Laws, The Three Laws of Thermodynamics, Newton’s Three Laws, Kepler’s Three Laws.

This is a General Rule of Laws, known as CalMeacham’s Third Law.

40

Don’t forget the Three Laws of Robotics.