Question

In the year 1637, Pierre de Fermat conjectured his infamous "Last Theorem,” and it wasn't until the year 1994—roughly 357 years later—that Andrew Wiles proved it after having worked on it for 7 years straight.

Fermat had a narcissistic tendency of often posing his "proven theorems" without their respective proofs, leaving it to other mathematicians to work the conjectures themselves. It was no different with his last theorem, where he noted:

a^n+b^n=c^n has no solutions with n > 2

Plus:

"I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."

To pro

Answer 1

To prove this theorem, Andrew Wiles had to make extensive use of modern mathematical tools such as the Shimura-Taniyama conjecture, elliptic curves and modular forms.

Do you guys think Fermat was centuries ahead of everyone else? Or perhaps we missed a much simpler way of proving his theorem? Or did he even prove it at all and just went with his wits?

I believe he never proved it but went with his logic and saw that it was unlikely for it to not be true.

Answer 2

Just curious to know what you guys think.

Answer 3

Honestly, I would agree. Men like Fermat have glorious minds but its sometimes forgotten that glorious logical reasoning is a part of that. I think he went out on a limb and decided that logically it was incredibly unlikely not to be true and it would be a very long time before he would be proven wrong if his claim was infact incorrect.

Answer 4

maybe there is a lost book of Fermat's proofs somewhere

Answer 5

It's impossible to find solutions to diophantine equations where the exponent is bigger than 2 maybe that's how he figured it?

Answer 6

Fermat definitely didn't know the Wiles proof and frankly it's difficult to believe that this problem--so worked over for hundreds of years--had a proof in the mathematics that Fermat knew that we haven't discovered.

Now, I imagine that sometime soon, someone will find some simplification of the Wiles proof as different the connections between the different branches of mathematics that Wiles used become clearer. But I'd eat my shorts if we ever found a proof that would have been accessible in the mathematics of Fermat's day.

Answer 7

Same here, but I wouldn't be impressed; mathematics is such a clever game, and it's very easy to make something more complicated than it should!

Answer 8

But we all know that the best proofs are those that are simple and elegant, such as the standard proof that the square root of 2 is irrational.

Wiles proof is anything but simple, but I understand from one of the maybe 200 people who can actually read it that it is elegant.

Answer 9

i checked fermat's facebook page and the proof is not their either.
if you have spare time and want to do something comprehensible, there are short proofs that there are no integer solutions to
\[x^4+y^4=z^4\]

Answer 10

I believe, like many good mathematicians, Fermat convinced himself he was right. Feeling more brilliant than his colleagues in this field of mathematics, Fermat decided that there was not a huge need for him to publish the proof. Rather, I believe, he expected people to take his word for it.

He could have also been in the position of Gauss with respect to non-Euclidean geometry. Gauss felt that the mathematical world was too dimwitted in comparison to himself to be able to conceive of non-Euclidean geometry. As a result, he worked out the entire system in private. Many years later, Bolyai (and the other guy I forget the name of) published his work on non-Euclidean geometry and was credited with its discovery. Slightly thereafter, Gauss sent a crass letter to Bolyai explaining, "I had already discovered all of this." Perhaps Fermat felt that his methods were too advanced for his time and did not want to be ridiculed or treated harshly by the mathematical community, just like Gauss.