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Fermat1
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In your opinion did Fermat have a proof for his theorem?
HallsofIvy said:Hey, if Fermat doesn't know!
My opinion (and it is only an opinion) is that what happened is what happens to all of us. Fermat thought that he had a simple proof, wrote a quick note to that effect, then went to bed. And discovered when he tried to carry out the proof, that he it did not work. That is supported by the fact that after he wrote that, he published proofs of the theorem for the cases n= 3 and 5. He wouldn't have done that if he had a proof for all n.
Fermat said:To my mind it doesn't really make sense that this kind of problem doesn't have a more elementary (less artificial I mean) solution. Having said that, if Fermat had a proof, he would have written it
Fermat's Theorem, also known as Fermat's Last Theorem, is a famous mathematical conjecture proposed by French mathematician Pierre de Fermat. The theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.
It is believed that Fermat had a proof for his theorem, as he wrote in a margin note in his copy of Arithmetica by Diophantus that he had a proof that was too long to fit in the margin. However, this proof has never been found and it is possible that Fermat made a mistake in his reasoning.
Fermat's Theorem is significant because it remained unproven for over 350 years, despite many mathematicians attempting to prove it. It was finally proven in 1994 by British mathematician Andrew Wiles, with the help of modern mathematical techniques.
The proof for Fermat's Theorem is quite complex and involves several different mathematical fields, including algebraic geometry, number theory, and modular forms. In simple terms, Wiles' proof relies on the Taniyama-Shimura conjecture, which states that every elliptic curve can be described by a modular form.
Fermat's Theorem has had a significant impact on mathematics, as its proof required the development of new mathematical techniques and theories. It also opened up new areas of research and has advanced our understanding of number theory and other mathematical fields. Additionally, it serves as a reminder of the importance of rigorous proof in mathematics.