A very old problem turns 20
plus.maths.org
plus.maths.org
If anyone is looking for a good book about the history of Fermat's Last Theorem I can thoroughly recommend Simon Singh's book about it [1], which is both accessible for those without a strong maths background (like myself) and paints a wonderful history of mathematics as well. In places it's more entertaining than fiction.
[1] http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-grea...
I'm an all round science nerd with a MSc in maths, my wife is an architect with a background in art history and graphic design and a aversion to all things mathematical, yet we both really liked book.
Note: I have felt deceived after reading Godel's Proof [1] since the authors claimed in the preface to have given an outline of the complete proof in the last chapter but did not (it was grossly incomplete). I am still indebted to the book though for teaching me how to think about pure mathematics.
[1] http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814...
It is a difficult proof, and some of the underlying basic concepts are already deep in their own right, deeper than will be covered in an advanced undergraduate degree.
There are some attempts at higher level overviews on the web, and they are slowly getting fleshed out.
The historical reasoning works like this. In Fermat's day, mathematical "publishing" was not submission of articles to peer-reviewed journals, but people interested in mathematics sending personal letters to one another. (One of the first mathematical journals, if I remember correctly, began as a collection of mathematicians' letters to one another, something like turning individual emails into an email list.) Fermat was a lawyer by occupation, and did mathematics in his free time. He corresponded with other mathematicians of his era, including Pascal. In all the surviving correspondence Fermat had with other mathematicians, there is no trace of a proof of "Fermat's Last Theorem," even though, as Colin pointed out, he wrote about simpler cases of that theorem.
The real historical reasoning comes in by considering dates of the primary sources. Fermat's note to himself in the margin of the printed Latin translation of Diophantus was written EARLY in Fermat's investigation of number theory. He read that book, taking notes in the margins as he read, and then he wrote for years afterward about interesting number theory problems. The book with Fermat's annotations was published after Fermat's death by Fermat's son, and that is when the marginal note became known to the world, but Fermat had plenty of time--and plenty of pieces of paper with wider margins--to write out a complete proof of his conjecture if he actually had one. So as essentially all mathematicians have concluded on mathematical grounds, and as all historians have concluded on historical grounds, most likely Fermat briefly thought he had a complete proof of his conjecture, which he then worked on, discovering a mistake in his reasoning. He proved a simpler case of the theorem, died, and then when his notes were published, other mathematicians worked on the full conjecture for 350 years before Wiles found the complete proof, based on advanced research by other mathematicians still alive today.
Google Books links to the historical background from mathematicians who write good popular books about mathematics:
5 years after that episode came out, Wiles showed his proof.
Quoting from Wikipedia:
In "The Royale", an episode (first aired 27 March 1989)
of Star Trek: The Next Generation, Captain Picard states
that the theorem had gone unsolved for 800 years. At the
end of the episode Captain Picard says,
"Like Fermat's theorem, it is a puzzle we
may never solve."
Wiles' proof was released five years after the particular
episode aired. This was subsequently mentioned in a Star
Trek: Deep Space Nine episode called "Facets" during June
1995 in which Jadzia Dax comments that one of her previous
hosts, Tobin Dax, had
"the most original approach to the proof
since Wiles over 300 years ago."
This reference was generally understood by fans to be a
retroactive continuity for "The Royale".http://www.youtube.com/watch?v=7FnXgprKgSE
It's a very moving piece that puts a lot of depth behind the 2d scribbling on paper that describes whole universes. Also a very humane look into the lives of mathematicians.
Reminds me so much of how people obsessed with a solution will give up any semblance of a social life to pursue it.
[1] http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-grea...
Edit: Just seen nicholassmith posted exactly the same link. Apologies for duplication
Would have been cooler if the proof was extremely simple to write out, but required outside-the-box thought process that evaded thousands of humans for centuries.
I hope the same thing does not happen to physics. Discoveries that lead to simple equations like e=mc^2 are so cool!
(Proving theorems is hard work!)
A good example is the Jordan curve theorem[1], which at first sight seems like a trivial problem but in effect requires a good deal of topology and analysis to even understand why it is so complicated in the first place.
I use this as an example of why "obvious" things aren't always true, need proving, and understanding their proofs can lead to useful insights.
[0] Edit: corrected "sphere" to "ball"
http://en.wikipedia.org/wiki/Alexander_horned_sphere (search for "jordan")
In essence, the "outside" gets very "tangled" and can't be smoothly converted into a "proper" exterior.
[0] S^2 = { (x,y,z) : , x, y, z, in R with x^2+y^2+z^2 = 1 }
I'd be interested to learn whether a linear version (straight pipes instead of curved/broken torus ones) also has the same topological properties. It is clearly fractal, and clearly also homotopy identical to S^2, but obviously the two ends are no longer interlocking, and I wonder if this is as crucial to the result as both 'ends' being fractal clearly is.
In any case, cool! thanks :)
It takes a while to get your head around what this really is.
That is, most people think of physical objects, where there is a minimum "planck length", and no fractal-ish structures with infinitely fine-scale structure.
It is still abstractly interesting that even 2-D fractals satisfy the Jordan Curve therorem, whie 3-D faractals do not satisfy the 3-D version of the conjecture.
> ... I find it unsatisfying that the solution to such a
> simply stated theorem requires 100s of pages of mathematics.
Wiles' proof is still comparatively new, it's still messy, it needs to be enhanced, clarified, expanded, and then "re-factored" to find the cleanest path to the result. The underlying arc of the proof is relatively easy to understand, it's the details that need the long, tedious, and careful arguments that take 100s of pages. One day it will be an obvious consequence of the full Modularity Theorem, discussed in honors projects of undergraduates.
> Would have been cooler if the proof was extremely simple to
> write out, but required outside-the-box thought process that
> evaded thousands of humans for centuries.
Only on the 18th century the series were discovered http://en.wikipedia.org/wiki/History_of_trigonometric_functi...
Correction : A Professor(Manindra Agrawal) who had been working on the problem for quite some time and 2 undergrads.
Having said that, it's quite possible that a more "elegant" proof will be discovered in the future. Maybe it requires further thinking outside the box -- it will more likely require new mathematics.
If you want really non-elegant proofs, take a look at computer-assisted proofs (e.g. proof of the four color theorem). These make the subject of this thread seem extremely elegant!
Think of this - I can use the number two and refer to it with just a single character - 2. However, properly stating all its properties and what it is starting from just the axioms of set theory, I'll need quite a lot of pages.
And that's how "lets store the number 2" goes from 1000 pages and ten hours of lecture to explain to CS grads, to some 5 minute Kahn academy video that any goof off the street can more or less understand.
There is no proof that any simpler explanation exists. Its quite possible no one will ever teach the proof of FLT to grade school kids. But if it ever happens it'll be like the analogy above.
If for example you simply accept everything up to and including the Taniyama–Shimura–Weil conjecture as true, then the proof of FLT gets pretty close to one line.
So someone Very familiar with the system is happy with one line of interactive shell running a trusted and well understood script. But if its weird, someone might demand to see the PBS batch script contents, or the innards of the perl, of if life and safety aerospace depended on it maybe they'd demand a down to the transistor formal proof of correctness (good luck with any non-trivial problem)
Or put another way I could teach you the equation for the current flow thru a MOS transistor vs gate voltage, maybe even run a simple lab until you're happy, and then say, well, there you go, and thats why my batch script works, and I have an elegant proof of how it works, beginning with this transistor equation, but the margin of my HN comment is too narrow to contain it, so...
E=mc^2 is just a statement of a relationship, it isn't a proof or a solution to a problem. Mathematical work in modern physics is in fact just as complicated and messy.
[1] http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
For the Ruffini theorem, http://math.stackexchange.com/questions/286927/why-cant-ther... states:
"The reason five is special is that the group A5 of all even permutations of 5 letters is the smallest non-abelian simple group".
An example from computer science that some may understand better: A monad is just a monoid in the category of endofunctors.
Both are short statements that one cannot grasp without prior knowledge, but that become simple once one has 'some' prior knowledge.
And in some sense, it is turtles all the way down. Even the statement 1+1=2, when studied deeply enough, requires huge amounts of prior knowledge (about 380 pages, and that excludes the definition of 'addition' (http://en.wikipedia.org/wiki/Principia_Mathematica#Quotation...) :-))
Some say that there are three kinds of theorems: false, trivial and not yet trivial.
Namely, I'm wondering if Fermat was actually lying and he just got lucky that a proof was eventually found.
It's technically possible that there's a proof that only requires the mathematics of that period, but since no one has found it yet it seems unlikely.
I'm told that if all rings of the form a+b.sqrt(-p) are unique factorization domains then there is a fairly simple proof[0], and it's plausible that Fermat momentarily forgot that this was not in fact the case. If he found the "proof" that uses this "fact" he may have got carried away and written his marginal note, then moments later done a face-palm, and not bothered to erase or correct that note.
[0] I do not speak from personal knowledge, and this may have been distorted. I believe the essence to be correct.
It should be noted that it had been proven true for a very wide collection of possible exponents. You can read about the piece-wise progress here:
http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_la...
As a piece of trivia, in the Horizon program[0] mentioned elsewhere there's a point where J.H.Conway[1] (inventor of the game of "Life[2]" (don't ever ask him about it!)) is being interviewed, and he uses the word "prime." The editors dubbed the word "number" over the top, feeling that the audience might be put off, and knowing that "number" was accurate enough for the purpose. I don't know if Conway was ever told - I'll try to remember to ask him next time I see him.
[0] http://www.youtube.com/watch?v=7FnXgprKgSE (roughly the 8 minute mark)
I'd link to the time in question myself in the present case, except that I listened from 7:00 to 9:00 and didn't find the thing you mentioned.
http://youtu.be/7FnXgprKgSE#t=7m55s
I didn't put the time marker on it because I wasn't able to check for sure, but I've confirmed it now.
I just can't resist showing why this is false. It's actually pretty simple.
Take p=5. Then our 'integers' in this ring are of the form a + b\*sqrt(-5) for integers a, b. Taking b = 0 we see that all the normal integers are part of this ring. So consider how 6 factorizes.
On the one hand we have 6 = 2 × 3... uncontroversial. But... due to the existence of new numbers in this expanded ring, we also get 6 = (1 + sqrt(-5)) × (1 - sqrt(-5)) . These are evidently two different factorizations... it suffices to show that both are prime factorizations under the modified definition of 'prime' we get when we allow factors of (a + b × sqrt(-5)). You can get there by noticing that any number of the form a + b × sqrt(-5) that factors into smaller such numbers must either be totally real (b=0), be totally imaginary (a=0), or have a larger absolute value than any number that could possibly multiply together with another number > 1 to get you 1 +/- sqrt(-5). This line of reasoning can show that 2, 3, (1 + sqrt(-5)), and (1 - sqrt(-5)) are all primes in this ring.
The real point is that assuming rings of that form are UFDs gives a relatively easy proof of FLT.