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 truly marvelous proof."

 

[master_coda]

There is no evidence that Femat recognized his proof was incomplete.

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]

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.

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]

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.

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 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..)

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

 

[master_coda]

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

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]

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.

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]

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

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 realize 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 realized 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]

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.

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!

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.

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.

 

[TenaliRaman]

Was Fermat too bold about her last theorem?

i never knew Fermat was female :D

-- AI

 

[TenaliRaman]

|_/\/\40!

:d

-- Ai

 

[mathwonk]

A remark, possibly of interest, involving confusing terminology. Wiles' background was not strictly in math of ellipses, but in math of elliptic curves. These curves may have arisen of the study of ellipses, in the time of the Bernoulli's say, in the following way, thus possibly giving rise to their name.

I.e. an ellipse is a non singular plane curve of degree 2, but an elliptic curve is a non singular plane curve of degree 3, or a curve "birationally equaivalent" to such a cubic.

Now as you know, if you have taught calculus and arc length, it is not so easy to compute the arclength of a parabola, and an ellipse is even harder, i.e. virtually impossible in elementary terms.

The integral you get when you try an ellipse if I recall is something like 1/sqrt(1+x^4). Now the plane curve defined by the reciprocal of this integral, y^2 = 1+x^4, is a singular plane quartic (the singularity is at infinity in the projective plane), but still birationally equivalent to a non singular plane cubic, hence is an "elliptic" curve.

So elliptic curves are one level more difficult than ellipses, which is actually quite a lot. And an elliptic curve is not an ellipse but the curve arising from the arclength integral for an ellipse. This beautiful story is told wonderfully well by C. L. Siegel, (using the lemniscate as his example, instead of an ellipse), in volume one of his Topics in complex function theory.

 

[Zurtex]

Sorry mathwonk, I knew that I just mis-typed.

 

[mathwonk]

Sorry to assume you did not know that.

I was a little too glib in my response though. I was not actually able to verify my claim that the arclength of an ellipse leads to the integral I asserted it did. It seems that is easier to check for the arclength of the lemniscate, which is probbaly why Siegel chose that one, or maybe for historical reasons, as he attributes the discovery to a Count Fagnano of Italy.
When i reviewed my work from calc last year, I found that I merely found a statement in my calculus book that asserted that the arclength integral for the ellipse reduced to an integral they called an "elliptic integral". so I am taking my calculus book's word for it that the arclength of an ellipse leads to an elliptic integral but I did not see how it worked out that way myself.

 

[ryokan]

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.

Gauss wrote: "Ich gestehe zwar, dass das Fermatsche Theorem als isolierter Satz fuer mich wenig Interesse hat..." (I confess that the Fermat theorem holds little interest for me as an isolated result...)
It was in a letter to Heinrich Olbers (May 1816)

 

[mathwonk]

I regret Gauss felt that way. Others with a longer time line, have bequeathed us the enormous heritage of knowledge, such as the theory of factorization, unique and otherwise by Kummer et al, growing out of their work on Fermat's problem, not to mention the big boost given to analytic number theory by Wiles' ideas.

So it would have been nice if Gauss had actually stated some of his problems for the rest of humanity to work on after his lifetime.

 

[ryokan]

... His last theorem

Was Fermat too bold about her last theorem?

i never knew Fermat was female :D

-- AI

No. He was a man with moustache.
Obviously, I made a mistake, when I wrote "her" to talk about his last theorem :blush: