Insights Fermat's Last Theorem

[elias001]

@fresh_42 it might be more than just a nice project. Imagine in the near future, anyone want fmto submit articles to a math journal is required to first formalise any theorems in their paper in Lean, and see if it passes muster according to Lean, then submit the article along with the Lean code. You aer going to tell professors who are teaching their students on how to do proofs using Lean that they should not be teaching their students to rely on it? Like I said, you should speak to the CS people on the Formal Verification of software and ask their opinions on the matter. It might or will be a matter of time before there are add on packages for proof checking into Maple mathematica and Matlab.

 

[elias001]

@fresh_42 here is the list of courses that have used Lean. Before you raised more objections, there are increasingly more engineering disciplines that are making use of algebraic geometry. Those fields are robotics, aresospace engineering, plasma fusion research and anything that has to do with systems and control along with coding theory and cryptography. Is not only the tech savy side of the math community that is interested. Uncle Sam's military has also funded research into this. Darpa is keeping their tabs on it. Actually, I won't be surprised there will be start ups in the valley if this technology has matured enough. There are start ups on translating the math you wrote on your kitchen napkin into LaTex.

Oh none of our opinions matter if this technology gets adopted, unless you are a field medalist or someone equally accomplished and uncontroversial.

 

[mathman100000]

Of course Fermat had a proof, why does everyone. Assume he made a mistake?he has never made a mistake, except for when he defines things as conjectures. He solved all even in an amazing proof, why would you think he couldn’t solve odd? If he were alive today he would be solving many open problems, he was just on another level. There infinite ways to prove each equation!

 

[jedishrfu]

History shows that the problem was far more intractable than Fermat let us believe. His comment was most likely a false epiphany, and once he began his proof, he realized there was a problem he couldn't overcome. He abandoned his approach but never bothered to retract the comment.

https://en.wikipedia.org/wiki/Fermat's_Last_Theorem

Andrew Wiles employed mathematical techniques that weren't known in Fermat's time, and even though he completed his proof, a problem arose that forced him to return to the drawing board, seeking help from one of his graduate students, Richard Taylor.

 

[fresh_42]

I just read about the ABC conjecture, which, if proven, would reduce FLT to one page:
Andrew Granville and Thomas J. Tucker, Nov. 2002, Notices of the AMS, Vol.49, No. 10, It’s As Easy As abc

 

[martinbn]

I just read about the ABC conjecture, which, if proven, would reduce FLT to one page:
Andrew Granville and Thomas J. Tucker, Nov. 2002, Notices of the AMS, Vol.49, No. 10, It’s As Easy As abc

Well, Wiles' proof of Taniyama Shimura reduces Fermat to zero pages.

 

[mathman100000]

When the next truly great mathmaticians comes along, he will also be able to reduce it to 1 page using techniques available to Fermat, just because our generations missed it, doesn’t mean it doesn’t exist? He solved for all even, I can’t imagine anyone else would have solve for all even, if his son hadn’t published it, as he was first to do infinite descent.

 

[jedishrfu]

@mathman100000, this is a personal opinion. While there is a non-zero percentage that this is the case, it is far more likely that Fermat was mistaken. He may have actually attempted to prove it, but ran into the same roadblocks as others did. It's something we will never know until his lost papers turn up.

https://en.wikipedia.org/wiki/Pierre_de_Fermat

There were other theorems that Fermat claimed to have proven but later mathematicians including Gauss thought he was exaggerating.

So you can go on believing he solved it with a short proof and maybe even attempt to rediscover what others missed but please let's move on to another more interesting topic until we have a peer reviewed paper of someone proving it using only mathematical methods known to Fermat.

 

[weirdoguy]

he has never made a mistake

Nonsense. Everyone make mistakes. Your whole comment is so out of touch with reality I almost think you are trolling.

 

[fresh_42]

While there is a non-zero percentage that this is the case, it is far more likely that Fermat was mistaken.

This is a very polite understatement, and a very optimistic and theoretical figure. Tens of thousands have tried; more than a dozen of them were great mathematicians, among others, Cauchy, Euler, Gauss, Kummer, Dirichlet, Legendre, etc. Fermat was quite talented, but I wouldn't list him as one of these names. And on top of it, several of them made mistakes or were wrong in their assessments. The chance that Fermat knew something Euler couldn't find is only theoretically above zero. I, on the other hand, think: no way.

 

[martinbn]

The statement that there is a nonzero chance that Fermat had a proof is just because we cannot be absolute sure about anything. But it means nothing. There is a nonzero chance that i will solve the Hodge conjecture tonight.

By the way I think that Gauss was not interested in FLT.

 

[fresh_42]

By the way I think that Gauss was not interested in FLT.

Depends on what you consider as being interested in.

The origins of the theory of modular forms go back to Carl Friedrich Gauß (1777-1855), who considered transformations of special modular forms under the modular group, the level in the context of his theory of the arithmetic-geometric mean in complex systems (1805).

Source: https://de.wikipedia.org/wiki/Modulform

Furthermore, he wrote an entire book about arithmetic (number theory), and $\mathbb{Z}[\mathrm{i}]$ which is used in some proofs is even named Gaussian ring in the German literature. According to Zachow, Gauss was the first, after Euler, who needed two articles, to prove the case completely for $n=3$ in a single paper.

Looking at the overview, it is noticeable that until the 1840s, Fermat's great theorem was only fully known for n = 3, 4, 5, and 7 through proofs by Fermat, L. Euler, or C. F. Gauss, L. Dirichlet/A. Legendre and G. Lamé (which covers at least two-thirds of the cases for n ≤ 100).

I would not call this not interested in.

There is a facsimile of the Disquisitiones Arithmeticae on the internet, but it cannot be searched electronically.

 

[martinbn]

Gauss wrote

"I confess that Fermat's Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can easily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our again several old ideas for a great extension of the theory of numbers. Of course, this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796-1798 when I created the principal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries."​

​

Which explains why I had the impression he had no interested in it. I think I had only seen the fist part of the quote.

 

[fresh_42]

I see no reason for shouting.

Yes, he didn't consider it important.

Even though he himself gave a proof for the case of cubes, Gauss did not hold the problem in such high esteem. On March 21, 1816, he wrote to Olbers about the recent mathematical contest of the Paris Academy on Fermat's last theorem: "I am very much obliged for your news concerning the Paris prize. But I confess that Fermat's theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."

Source: Paulo Ribenboim,13 Lectures onFermat's Last Theorem, Springer 1979

Nevertheless, he gave some results for $n=3$ (the solution with the help of complex numbers, and a theorem for finite fields), and helped to pave the way with his contributions to arithmetic. I think Gauss's opinion summarizes it: FLT itself is not very important to number theory, but the attempts to solve it have been.

 

[martinbn]

I see no reason for shouting.

I am not shouting! If you mean the boldface, i cannot edit it for some reason.

 

[weirdoguy]

Sorry for OT, but it always strucks me when brought up: I don't know why but in polish internet boldface is not considered shouting, caps lock is. Weird.