Was Fermat's last theorem really as difficult as it seemed?

[Rothiemurchus]

How was Fermat's last theorem proved and why did it take so long to solve this problem?
Is there a shorter,more elegant solution waiting to be found?Can the methods of calculus be used to solve this problem?

 

[matt grime]

How was Fermat's last theorem proved

read the book. (and it is too hard to explain here, though basically it asserts that some canonical bijection between two sets of objects was written down).

and why did it take so long to solve this problem?

because it is very hard to even think of the method to attack this problem, never mind find the correct argument using that method.

Is there a shorter,more elegant solution waiting to be found?

probably not or it would have been found by now.

Can the methods of calculus be used to solve this problem?

depends what you call calculus, but I'll plump for 'no': you need to know a lot more maths than the average PhD in maths understands to appreciate the proof, and that certainly goes beyond whatever you mean by 'the methods of calculus.'

 

[Werg22]

Calculus or not, I think at the levels at which to prove is read, branches superpose each other constantly and hence the label "calculus" wouldn't mean anything.

 

[popi]

I was also wondering whether there is a simpler solution to the problem. Afterall, Fermat claimed he had the proof and I can't imagine his proof being even remotely as compliated as that of prof. Wiles (the Taniyama-Shimura conjecture and stuff...)

Was Fermat just bragging (which I doubt) or did he have a sort of ˝approximate˝ proof?

 

[robert Ihnot]

I was also wondering whether there is a simpler solution to the problem. Afterall, Fermat claimed he had the proof and I can't imagine his proof being even remotely as compliated as that of prof. Wiles (the Taniyama-Shimura conjecture and stuff...)

Was Fermat just bragging (which I doubt) or did he have a sort of ˝approximate˝ proof?

Well, when I went to school before Wiles' proof, that matter was given over to much conjecture and there was no clear answer. Some believe he made a mistake about unique factorization, as did Kummer. Many, many amateurs have been absolutely sure that Fermat had his proof, and have used that premise to believe that some simple idea is just laying around waiting to be found by the lucky one. But that has not been shown to work out.

There was in 1908 the Walfskehi prize of 100,000 marks for a correct solution. But that largely disappeared in the collapse of the mark, but was restored in present times and collected finally by Wiles. It was worth $50,000.

Walfskehi was an interesting story in himself. He was going to commit suicide, and arranged everything in his office in a very methodical way. He set the exact time, but suddenly he thought he had found a solution to Fermat's Last Theorem, and he worked long and hard at that. But by the time he found his mistake, the hour of his death passed; and, of course, as methodical as he was, he had to drop the whole matter. He proved so grateful about that that he created the prize.

 

[HallsofIvy]

No, Fermat was not "bragging" (after all, he wrote his comment about having a simple proof in the margin of a book only he read- it was not even discovered until after his death). What happened to Fermat was the what happens to mathematicians regularly- he thought he saw a way to give a very general proof, wrote down a comment to remind himself, and then when he looked more closely, saw that it die not work.