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

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In summary: Unfortunately, he never wrote down the proof that he thought he had, so we will never know.In summary, Fermat's last theorem was proved by Andrew Wiles using a canonical bijection between two sets of objects. It took a long time to solve because it was a difficult problem with no clear method to approach it. There is likely no shorter or more elegant solution waiting to be found, and the methods of calculus would not be sufficient to solve it. Fermat's claim of having a proof was not just bragging, but he likely made a mistake and never wrote down the proof he thought he had.
  • #1
Rothiemurchus
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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?
 
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  • #2
Rothiemurchus said:
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.'
 
  • #3
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.
 
  • #4
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...:confused:)

Was Fermat just bragging (which I doubt) or did he have a sort of ˝approximate˝ proof?
 
  • #5
popi said:
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...:confused:)

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.
 
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  • #6
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.
 
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What is Fermat's last theorem?

Fermat's last theorem is a famous mathematical problem that was first proposed by French mathematician Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.

Has Fermat's last theorem been solved?

Yes, Fermat's last theorem was finally proven by English mathematician Andrew Wiles in 1994 after more than 350 years of attempts by countless mathematicians. Wiles' proof uses advanced mathematical concepts such as elliptic curves and modular forms.

Why is Fermat's last theorem significant?

Fermat's last theorem is significant because it is one of the most famous and long-standing unsolved problems in mathematics. It has also inspired the development of new mathematical techniques and theories in order to find a solution.

What are some real-world applications of Fermat's last theorem?

Fermat's last theorem does not have any direct real-world applications. However, the mathematical concepts and techniques used to solve it have had significant impacts in other areas such as cryptography, number theory, and algebraic geometry.

Are there any other similar unsolved problems in mathematics?

Yes, there are many other unsolved problems in mathematics, such as the Riemann hypothesis, Goldbach's conjecture, and the Collatz conjecture. These problems continue to challenge mathematicians and inspire new developments in the field.

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