Is it possible to explain the solution to Fermats Theorum in plain english for a complete mathematical moron? And maybe what use the formula he is solving is in real life?
I am reading the Girl Who trilogy, and a thread through the second and third book is a solution to the theorum. Unfortunately my ability with math is about elementary school level [thanks to a fucking assbitch of a math teacher in 4th grade, I am phobic about math, and pretty much incapable of doing more than picking up a calculator at this time. I managed to force myself to memorize shit in school long enough to take tests, but that is literally it.]
All I really know is I managed to understand about 3 paragraphs in the wiki on the theorum
No, the Theorem is dead easy to explain, but the proof turns out to be a monster - it’s highly doubtful that Fermat did indeed have a proof, although “this margin is too small to contain it” was certainly the understatement of the century,
Fermat had several theorms that bore his name. The most famous one is Fermat’s last theorem, but I have trouble believing that the proof of that theorem is contained in a popular fiction work, as its not the sort of thing even most mathematicians are likely to be able to understand without specialized knowledge.
wiki: no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. (the n above should be superscript-a to the nth, b to the nth…)
It is a thread in the Girl Who…but just mentions the cubes of the three variables.
If you think understanding Fermat’s last theorem and the 358 year process of proving it will increase your enjoyment of the books, you might be disappointed. It’s more likely to annoy you that Larsson states the theorem incorrectly, misrepresents Wiles work in proving it, and has his character solve a special that was proved by Euler in 1770.
It isn’t that I think it will help me enjoy the book, it is more that I find it odd [?] that you can have a formula that has no solution[?]
not making myself clear.
Of what use is a formula with no apparent use or solution? With my intense lack of math I am not sure how to really explain. To me it seems like you are going
apples multiplied by oranges equals watermelons
something that has strange greeky letters and numbers and absolutely no apparent use whatsoever because a whole bunch of math geeks say it has no solution.
First of all it’s an equation that does have solutions, those solutions just don’t happen to be triplets of positive integers.
Secondly, it’s an equation that looks suspiciously like an equation that has an infinite number of solutions that are triplets of positive integers, the Pythagorean theorem.
This is the kind of thing that makes a mathematician ask “why?” For some problems the solution is obvious. Some take quite a bit of cleverness. And some take some of the best mathematical minds three and a half centuries.
“What is the point of an equation with no solution?” is not a question mathematicians are overly concerned with. “Why doesn’t this equation have a solution?” can be an interesting exercise, a dissertation topic, or a life’s work.
Well it can still be a useful or interesting observation before it’s been proved to hold for all cases.
e.g. If a theory holds for the first billion positive integers, then fails on 1 billion + 1…that in itself is interesting*. Why does it fail at that point?
<hijack>
One thing I wonder is: Has there ever been a mathematical theory where an intuitive proof in natural language was possible first, and a proof using mathematical axioms came significantly later? </hijack>
Unless the theory is “All integers are <= 1 billion”
This book, in my opinion, does a very good job of explaining the proof accessibly, requiring little more than high school math. It’s also written in a very interesting and historical way, chronicling the various attempts to prove the theorem by different people, the steps and missteps along the way.
Maybe as a hobby, but nothing concrete [or profitable other than books to a seriously niche audience]
I suppose you could lock a criminal mathematician in a cell and make him work on unsolvable stuff for food [and cruel amusement for the guards] Very Gilbert and Sullivan
That was my problem in school. I spent so many years in art classes I can draw anything, proving it is entirely different. :smack:
I will see if I can crank out some amazon mechanical turk work and pick it up.
If you read the Wikipedia entry on Fermat’s Last Thereom, you will learn as much as most mathematicians know about the subject:
Really, most mathematicians (at least those who were out of grad school when the proof was announced in 1993, as I was) know no more (and probably less) about the proof than what you will find in the entry. They know what the theorem is about, the history of partial answers to it and mistaken proofs of it, and a little about the mathematical subfields that are used in the proof. Most of them don’t have the free time for the weeks or months of work that it would take to learn the material in those subfields (which are often quite far from their own subfields) to comprehend the proof.
For a non-mathematician, I think the following explanation is sufficient:
It’s easy to find cases of (a,b,c), where a, b, and c are positive integers, where (a^2) + (b^2) = (c^2).
Here are all the cases where a, b, and c are all below 100:
An obvious question then is if there are cases where (a^3) + (b^3) = (c^3). Today, with computers, you could try enormous amounts of (a,b,c) examples and find that none of them work. You could then try to find other solutions to the general equation (a^n) + (b^n) = (c^n) for other positive integers n. With computers today, you could try lots of cases and find no solution to this equation.
The question that arises then is if it’s true that for positive integers (a,b,c,n) if there are any solutions to the equation (a^n) + (b^n) = (c^n) where n is anything other than 1 or 2. For three hundred fifty or so years mathematicians slowly found proofs that the equation couldn’t be solved for various values of n. Finally Wiles (and Taylor) were able to show that it’s not true for any value of n other than 1 or 2.
To be fair, for the curves that you could actually draw, “well… look at it!” is about as tricky as the proof is. It’s when you start dealing with fractal curves and the like that things get harder.
Thing is, a[sup]3[/sup] + b[sup]3[/sup] = c[sup]3[/sup] has lots of nice tidy solutions if you’re not fussed about round numbers. 1[sup]3[/sup] + 2[sup]3[/sup] = 9, and 9 does have a cube root… just not a round one. So then you start wondering “Well, hang on, is it possible to find any numbers that it works for, the way 3[sup]2[/sup] + 4[sup]2[/sup] = 5[sup]2[/sup]?” And the answer turns out to be “No, given the infinite number of whole numbers you could plug in, none of them work” and then you find it’s true of a[sup]4[/sup], a[sup]5[/sup] and any higher powers you can imagine. But the proof… I may never be competent to understand it.
As for that margin note: Fermat did, in fact, come up with a method of proof that works for n=3 and n=4. When he was first developing that method, he probably thought that the method would generalize to all values of n. And so, he scribbled in the margin of one of his books that he had found a method to do that. Later, of course, he turned out to be mistaken: His method worked for 3 and 4, but not for any higher power. But it’s not like he ever published the initial claim: He never expected anyone else to read his margin scribblings (if he even remembered he had scribbled it), so there was no point in issuing an erratum to it.
Fermat’s Last Theorem is an example of a long-unsolved mathematical problem. It’s like the mathematicians’ version of 14 k of g in a f p d. It’s the kind of thing that mathematicians work on, not because it’s useful or has applications to the real world, but because dammit, they want to know the answer to the question. It’s the Search for Truth.
Mathematicians study questions like FLT for the same reason theologians study god. With the exception that had a mathematician proved that the world had to end on May 21, it would have ended on that date. If the premises and proof were correct.
But as for the OP, I do not expect that I could ever understand the proof, unless I spent a year of concentrated study on it. Unless a substantial simplification were discovered (not unlikely, to be sure, most proofs do eventually get simplified) very few mathematicians even will ever understand it.
Pretty much all mathematics was developed this way. First, people play around thinking abstractly about whatever pre-formal intuitive concepts they like. Then, later, with whatever discoveries they’ve made in that process as a roadmap, some choose to concentrate on formulating the relevant axioms to support the existing large body of work’s formalization (and further formal development). By and large, pre-formal mathematics drives formal axiomatization, not the other way around.
Well, you don’t get paid for thinking about math that you can’t do. You get paid for thinking about math that you can do. It’s just that there’s a great awful lot of math, so much that no single individual thinks about all of it.
