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A very old problem turns 20 (maths.org)
99 points by ColinWright on June 24, 2013 | hide | past | favorite | 56 comments

The history of the Theorem and the attempts to crack it, and how Wiles finally got there is fantastic, and if you've not come across it in detail then I'd definitely recommend finding out more about it.

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'd 2nd that. I've read two other of his books, The Code Book and Big Bang, in which he gives cryptography and physics similar treatment.

After reading the book, would I have some understanding of the actual proof, or learn mostly the historical context around it?

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...

A popular science book about Fermat's Last Theorem will certainly not give you any kind of real understanding of the proof. It will give you some historical context.

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.

Great book, no matter your level of math knowledge or interest.

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.

Somewhat off topic, but I was somewhat surprised that architect is a viable career for someone with an aversion to all things mathematical. Does she still need math, she just doesn't enjoy it, or does the modeling software these days take care of all the calculations necessary?

The role of the architect in Sweden is very different from the role of the architect in North America. In Sweden an architect deals with the planning and design phase and is not involved with the construction phase and has no practical or legal responsibilities with regards to the actual building process. All that is handled by a construction engineer.

Interesting. I would have thought at least some math would be required during the planning/design phase to ensure the design is physically stable (e.g. load calculations to ensure spans are supported well enough). Thanks for the info.

Most houses are designed with standard, well understood materials and well within industry standard tolerances, so there is no need to do any heavy duty load calculations at every step. 97% of the time plugging some numbers into the standard formulas is plenty enough for a sanity check. If you're doing something really weird you sit down in collaboration with your engineers.

I read the book as a kid (I'm guessing 15), and while I probably understood little of it, I read the whole thing. It was definitely inspiring to learn Wiles' slow persistence/progress, and the never-would've-guessed kind of connections that come up in maths.

Some comments ask whether Pierre de Fermat may have had a proof of his conjecture (which I now call the Fermat-Wiles Theorem) using the mathematics of his day. Agreeing with Colin, on historical grounds, I think Fermat did NOT have a proof of his famous conjecture.

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:



Fun fact: In Star Trek the Next Generation, Picard mentions that Fermat's Last Theorem had gone unsolved for 800 years.

5 years after that episode came out, Wiles showed his proof.

Season 2, Episode 12, "The Royale." This was subsequently tangentially referenced in an episode of "Deep Space Nine," which means "explanations" about Wiles never being born in the timelines are wrong.

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".

in Picard's parallel universe Wiles wasn't born.

See my answer elsewhere explaining why this is not true.

Here: https://news.ycombinator.com/item?id=5932237

There's a lovely documentary from the BBC/Horizon on Andrew Wiles' accomplishment. Also ran on PBS as the "The Proof".


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.

Here is the book[1] that goes with that documentary. It is a really good read and not too technical at all.

[1] http://www.amazon.co.uk/Fermats-Last-Theorem-confounded-grea...

Edit: Just seen nicholassmith posted exactly the same link. Apologies for duplication

I'm not a mathematician, but I find it unsatisfying that the solution to such a simply stated theorem requires 100s of pages of mathematics.

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!

  > ... I find it unsatisfying that the solution to such a
  > simply stated theorem requires 100s of pages of mathematics.
The mathematics involved is getting easier and easier to understand as more and more work is being put into it. Although Wiles originally only proved the underlying result for semi-stable elliptic curves (which was enough for FLT), the full modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture) is now proven, and the field continues to grow. I have little doubt that in 100 years or so this will be within the scope of later stage undergraduate work, just as many results in number theory (and other areas) become more and more accessible as we come to understand them properly.

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.
In some sense, perhaps. That's what happened with the proof that primality testing is in P. Three undergrads produced an elementary argument that had eluded people for centuries. And yet it's provided no really new insights into anything, and seems to be effectively a dead end. In that sense it's much more exciting to have genuinely new work that wasn't just a simple argument overlooked for centuries.

Yeas, I find it difficult to believe that it'll get easier, but then again, calculating the sine for an arbitrary number was once impossible, later very difficult (and only in the realm of 'pros') and today it's a couple of button presses away.

Actually, it's pretty easy to get a few places of accuracy purely by hand using the first couple of terms of the right series, and a few double (and triple) angle formulas.

Ah yes, but the series came much later in history

Only on the 18th century the series were discovered http://en.wikipedia.org/wiki/History_of_trigonometric_functi...

> Three undergrads

Correction : A Professor(Manindra Agrawal) who had been working on the problem for quite some time and 2 undergrads.

Well, why exactly would you think that a simple to state problem should have a simple solution? It is of course false, as demonstrated for example by the insolvability of equations of degree higher than 5 [1] and by countless other examples, but I don't see a reason why it would even be a reasonable heuristic. If you spent just a few hours trying to think of a proof of the Fermats theorem you would very quickly see a lot of good reasons why it is difficult and why an elementary solution most likely does not exist.

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

As others said, once you have sufficient terminology, the proof is simple.

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.

What often happens is the first version of a proof is long and arduous, but the new ideas that it spawns slowly circle back so that someone else can come up with a much neater proof, it just takes a while.

(Proving theorems is hard work!)

In other cases, a very complex problem can be stated very simply, especially to a lay person.

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.

[1] https://en.wikipedia.org/wiki/Jordan%27s_theorem

There is an obvious enhancement of the Jordan Curve Theorem, which is to say that the interior is effectively a disk, and the exterior is effectively a plane with a hole. There is then an obvious 3D version, with a (topological) sphere separating space into a topological ball[0] and a topological R^3 with a hole. This enhanced version, although natural and obvious, is false.

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"

(I think) this is a case were lay intuition assumes objects are piece-wise differentiable, but the formal math definitions don't.

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.

Ooh, OK, you've tickled my interest. Can you provide an explanation (or a link to something similar) with the disproof of the 3D version?

In three dimensions there is an embedding of S^2 (which is the surface of a sphere[0]) into R^3 such that the exterior is not homeomorphic to R^3 with B^3 missing.

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 }

Hmm, so because part of the surface is a fractal, it can't be simply connected?

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 :)

The surface is simply connected, it's the outside that ends up not being simply connected. This works equally well with piece-wise linear embeddings. And the current version of the Alexander Horned Sphere is not actually interlocking - the embedding is contractable back to S^2.

It takes a while to get your head around what this really is.

OK, "proximate" rather than interlocking. I did understand the geometry of it, I just chose the wrong word. Thanks for your explanations! :)

I think that's largely because mathematics abstracts more than is obvious. My favourite example is Hilbert's basis theorem, which looks like it's not even a question until you realise how general the notion of "space" that it applies to.

Right, same reason that the Banach-Tarski paradox is intuitively baffling. It relies on fractal-structures that require uncountably-infinite fine detail.

What I find interesting is that these proofs aren't one liners. To me, I intuitively presume that mathematical proofs should be one liners because they are explaining fundamental truths of nature; this universe and beyond.

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!

The proof is a one liner, if you take enough of the background theorems as given. The reason the proof is so long is that Wiles had to prove and build upon a bunch of other theorems to get to a point where proving FLT was possible.

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.

The best explanation I can give that will probably resonate with HN folks is I did a big data import this morning... was it a one-liner because "vlm@something:~$ submit.pbs.job.sh" is one line, or is it a ten or so lines of batch submission stuff, or is it one line running a perl script that rips apart one file format, cleans the data a bit, and shoves it into a DB or is it the one line of perl code, or one line of C, or one assembly line, or one byte/word machine language opcode, or one microcode operation, or one transistor?

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...

It can be a one-liner, but you'll need more language. That is, you'll need terms that encode mathematical truths that currently can only be expressed as several pages.

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.

Maybe a good way to resonate the situation with HN would be you wanna store the number 2. Well, if you only know and are permitted to use classic IEEE 754 floating point then this is going to be a really long story. But once someone "invents" binary integers and its widely accepted that you're "allowed" to assume everyone understands them, then to store the number 2, you just squirt out a 8-bit binary integer word 00000010 and call it good.

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.

It should be fairly simple to calculate the simultaneous gravity effects of 3 or 4 planets on each other, then plot their courses. That simple problem has not been solved yet!!


I've occasionally caught myself wondering whether there is a completely different proof that could have been discovered using 17th century mathematics. Or is it the case that the maths used by Andrew Wiles would have been understandable to Fermat?

Namely, I'm wondering if Fermat was actually lying and he just got lucky that a proof was eventually found.

It is almost certain that Fermat did not have a proof. He did have (effectively) a proof for the case n=4, but it seems he worked for some time without success on a proof for n=3. It seems unlikely he would do so if he really did have a general proof.

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.

>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

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.

So you have proven that a ring of that form is not necessarily a UFD by giving a specific case where it isn't, and proving it.

The real point is that assuming rings of that form are UFDs gives a relatively easy proof of FLT.

Were Mathematicians generally convinced that a proof did exist and it was just a matter of time? Or was there a not-insignificant belief that it could be false?

The mathematicians I know and who had expressed an opinion generally believed it was almost certain to be true, but that the techniques and machinery were not yet available to tackle the problem. There was genuine astonishment when Wiles produced his proof, but no real disbelief, as might be expected had people felt it not to be true.

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:


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)

[1] http://en.wikipedia.org/wiki/John_Horton_Conway

[2] http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

N.B. If you click on "Share" at YouTube, it includes a box and a field with the time in the current video. If you check the box, it gives you a direct link to that time. (The format is just http://youtu.be/7FnXgprKgSE?t=9m13s, so you can also do it directly.)

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.

It's between 8m03s or 8m04s, and it's very hard to tell - it's very well done. If Simon Singh hadn't told me I would never have known. Here's the section:


I didn't put the time marker on it because I wasn't able to check for sure, but I've confirmed it now.

More likely Fermat was just mistaken. There have been numerous flawed proofs over the years --- most likely Fermat stumbled on one of those, realized the error (hence, never published), but never bothered to correct the note he'd left in the margins on of one his books.

Fermat definitely couldn't have understood Wiles' proof in terms of the mathematics of his time, since it relies on a lot of theory that wasn't available then.

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.

Not lying, but probably thought quickly of a proof that later was revealed to not be a correct one (not sure Fermat got to this point)

That's a good point, for some reason it didn't occur to me that he might simply have been wrong...

I agree; if he didn't write it down (and test it), it was probably a simple one that was subtly wrong.

I don't have a copy of the book in front of me, but I seem to recall Simon Singh's book talked about simple, but flawed attempted proof that was found in the 19th century, that might have been what Fermat found.

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