MHB Fermat's Theorem: Did Fermat Have a Proof?

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The discussion centers on whether Fermat had a proof for his famous theorem. Participants express skepticism that Fermat possessed a valid proof for all integers, suggesting he may have believed he had one but later realized it was flawed. The mention of Fermat publishing proofs for specific cases (n=3 and n=5) supports the idea that he did not have a general proof. Additionally, the consensus is that if a simpler proof existed, it would likely have been discovered by the many mathematicians who explored the theorem over the centuries. Overall, the discussion leans towards the belief that Fermat did not have a proof that would withstand scrutiny.
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In your opinion did Fermat have a proof for his theorem?
 
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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.
 
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.

I share your opinion on this. :D
 
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 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

Considering the monumental giants who tackled the problem during the centuries it remained unresolved, I tend to think that if a more elementary proof of the theorem existed, it very likely would have been found. :D
 
Definitely agree he did not have a proof that would hold up. I would be very curious to see what it was though.

There was an even better video on the subject I saw once, but this one is also interesting.

 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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