Was Fermat too bold about her last theorem?

ryokan

I have seen that there were two threads on the Fermat's last theorem.
In basis to the complex work which was developped in the recent years, culminating in the Andrew Wiles' work, is credible that Fermat had a "demonstrationem mirabilem" for such theorem?

I remember his quote:

"Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.
Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caparet"
(It is impossible for a cube to be written as the sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain)

Zurtex

Most mathematicians accept that there was probably some flaw in Fermat's proof quite simply because it would make all other mathematicians, who have tried to solve this, look fairly stupid

master_coda

Fermat later went on to prove a special case of the theorem after writing his initial note. If Fermat actually had a complete proof for the theorem in general, then it's highly unlikely he would have went to so much trouble to prove special cases of it. It's much more likely that he later on realized that his proof was incomplete.

robert Ihnot

There is no evidence that Femat recognized his proof was incomplete. Thought he did recognize that he was not certain about the primality of (2^2^p)+1. Fermat as a general rule never published any proofs, but he did make exception with "Fermat's Last Theorem," i.e. the case for n=4. He gave no proof of "Pell's equation," nor any proof that every integer is the sum of three triangle numbers, 4 squares, et. These are very significient theorems. Along with Pascal he developed the theory of probability, offering Pascal the complement, "I see that you can keep up with me in this." But when he turned to number theory like (2^2^p)+l, Pascal assured Fermat that he had no understanding at all.

Fermat was not a professional mathematician. His was not a case of "publish or perish." He ranks probably as the greatest of number theorists, and his (assumed) theorems were offered only as challanges.

As for the Last Theorem, he never mentioned it in correspondence and it was only discovered written in the margin of a book after his death. This is not a bold approach, but he did assert in that margin that, "I have discovered a truely marvelous proof."

master_coda

You could argue that there is not 100% conclusive evidence, but there certainly is very strong evidence. Fermat originally wrote his now-famous comment long before he provided a proof of a special case of the theorem (and discussed proofs of several other special cases of the theorem). That is a great deal of time to waste when you have a "marvelous" proof of a much stronger result.

The evidence is not the fact that he did not publish the general proof, it's that he went to so much trouble to produce a weaker result after he thought he had a marvelous proof of a stronger one. This suggests that he recognized that his stronger proof was flawed.

Zurtex

Perhaps, but one of the things about Fermat is he didn't like revealing his proofs. He often wrote to famous mathematicians giving them problems he has solved and not providing a proof just to see if they could solve it themselves. Also it was very much a time of maths only just coming out of secrecy, long before mathematicians had kept their work secret so only they would be able to solve particular problems.

matt grime

And the actual rigour of proofs was not what a modern mathematician is expected to provide.

master_coda

Yes, but this still misses the point. Fermat devoted time later in his life to proving a special case of FLT. The issue isn't "if he had a solution, why didn't he write it down or tell anyone about it". The issue is "why did he devote so much time to proving something if he already found a solution years before".

arildno

I agree with mastercoda:
The sentence about the "Last Theorem" is, from what I know, written down in what is generally regarded as Fermat's first excursion into math's:
A book on Diophantine equations, I believe.

Since he never referred to the Last Theorem in any of his correspondence (again, from what I know), isn't it more probable that the famed sentence is just an enthusiastic comment from a novice, who later saw the flaw in his own proof?
(With the ink they used then, it wouldn't be easy for him to erase it later on..)

Zurtex

I thought it referred to the fact it was the last theorem that was still either be proven or disproved.

master_coda

Yes, that is why it is called the last theorem.

Nenad

it has been prooven, by a British Mathematician back in like 1998. His name was Andrew Wiles, and he got the complete thing. He didnt use the kind of math that Fermat would have used, but he proved the theorem.

master_coda

I assumed everyone knew that, since it was mentioned in the opening post. It was certainly the last of his theorems to be proved, by far, but it has been proven.

mathwonk

The cool thing to me about the proof was the beautiful and rather elementary idea, that started Wiles off on it. As you probably know, there is a very interesting type of equation in number theory and algebraic geometry called an elliptic curve, defined by choosing three numbers, say a,b,c, (subject to a genericity condition) and forming the equation y^2 = (x-a)(x-b)(x-c).


Then somehow Gerhard Frey got the idea that if the numbers r,s,t satisfying Fermat's equation, x^n + y^n = z^n, could not exist, then probably somehting else made from those three numbers also should not exist, like an elliptic curve.

Indeed it turned out that the elliptic curve formed from three numbers, slightly different from but related to the three putative solutions of fermat's equation, would give an elliptic curve which did not come from a "modular form". This fact was proved by Ken Ribet. Then Wiles proved almost all elliptic curves did come from modular forms, and the ballgame was over.

So the technical prowess of Wiles is fantastic, that none of us could have done, but the idea of Frey, is something it is tempting to daydream that we might have thought of ourselves with enough imagination.

To me it is inconceivable that Fermat had anything like a real proof, or even a good idea, after all that has gone under the bridge about this equation in the intervening 300 years.

the proof for n=4 is very elementary, and uses only two facts:
1) we know all positive integer solutions of x^2 + y^2 = z^2.

2) None of those solutions has the property that every solution is a square.

This can be explained in detail to beginning freshmen (I have done it).

Even the case n=3 is a good bit harder, attributed perhaps to Euler. Is Fermat even believed to have known that case?

As I have asserted before however, solvinghard problems is not as rare as coming up with good interesting challenging problems, so Fermat deserves even more credit for that.

The other biggie of that nature is I suppose the Riemann hypothesis, that all zeroes of the zeta function have real part (?) equal to 1/2.

As far as I know no one has any good idea as to how to attack that one, although there has been over 150 years of work on it. Ideas like trying to show some certain percent of the zeroes must lie on that line (like 38%), are fun, and lead to publications and PhD's, and are learning exercises, but of course can never give the result.

So all daydreamers who think Frey's idea was not so unthinkable, are invited to try their creativity out on this one.

master_coda

It's know that Fermat discussed proofs of the n=3 case, but it's not certain if he actually knew a (complete) proof of it.

The first proper proof of it is generally attributed to Euler, athough the actual proof he produced was incorrect. However I believe the first proper proof of it is just a variation of Euler's proof and that's why Euler is credited with it anyway.

Zurtex

I very much like the way Whiles tackled the whole problem, with a good background in the mathematics of ellipses he was able to prove the Taniyama - Shimura conjecture. I realise the idea was not his but even so when you first look at the problem xⁿ + yⁿ = zⁿ, your mind does not link it to ellipses and modular forms. I do believe that many values of n were proven however there were problems proving certain prime values of n, even when computers came along mathematicians realised they could carry on proving for values of n (and I do believe it got up to somewhere like 5'000'000'000) but it didn't get close to tackling the problem.

shmoe

I can't find the exact quote, but Gauss said something along the lines that he wasn't interested in working on Fermat's last theorem because he could easily come up with many problems that were very simple to state yet would never be solved one way or another in his lifetime. Challenging problems that are actually solvable and interesting are what's really difficult to come up with!

It depends on how you define "good", but it seems there's always some good looking ideas on how to tackle RH. There's quite a bit of buzz the past few years relating to the connections between zeta and the Gaussian Unitary Ensemble. Though relating the zeros to eigenvalues isn't a new idea (Hilbert & Polya years ago), recent work (last 20 years count as recent?) with computers allowed sufficient statistics on zeta to be gathered and compared with statistics for the GUE (and of course the whole "pair-correlation" correlation from the early 70's). There has been some definite good out of these ideas, with promising conjectures on the higher moments of zeta being made pretty recently. Course, it's not clear how to prove these conjectures, but at least they've been made.