Insights Fermat's Last Theorem

[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.