Have I proved some part of Fermat's last theorem?

[fenstip]

Have I proved Fermat last theorem?

X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z.

x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat theorem, in which n=4.

X^8 + Y^8 != Z^8 // replacing x,y,z with X,Y,Z.

In the same way, you can have n=16, 32, 64...

 

[martinbn]

Have I proved Fermat last theorem?

No, you've just made a well known observation.

X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z.

x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat theorem, in which n=4.

X^8 + Y^8 != Z^8 // replacing x,y,z with X,Y,Z.

In the same way, you can have n=16, 32, 64...

What about others? Say n=5.

 

[berkeman]

Thread closed temporarily for Moderation...

 

[fresh_42]

Thread re-opend.

 

[fresh_42]

It is sufficient to prove Fermat's last theorem for odd prime numbers, and as far as I know, it is believed that Fermat had a proof for three in mind. He only wrote down a sketch of a proof for four so we cannot know for sure what he knew and what he did not know. The Pythagorean triples which solve the quadratic case have been known since ancient times. Euler has definitely proven the cases three and four.

 

[mathwonk]

Here is a blog where Fermat's proof sketch for n=4 is elaborated.
http://fermatslasttheorem.blogspot.com/2005/05/fermats-one-proof.html

Another proof is given in Part Two, chapter XIII of Euler's wonderful introduction to algebra, Elements of Algebra. p.361 of this copy:
https://archive.org/details/ElementsOfAlgebraLeonhardEuler2015/page/360/mode/2up

 

[fenstip]

No, you've just made a well known observation.

What about others? Say n=5.

N=3,4,5 has been proved hundreds of years ago, if let them mix and mix in my proof can get N more dense, but far away from complete dense. Only Andrew Wiles has the full proof.

 

[fenstip]

It is sufficient to prove Fermat's last theorem for odd prime numbers, and as far as I know, it is believed that Fermat had a proof for three in mind. He only wrote down a sketch of a proof for four so we cannot know for sure what he knew and what he did not know. The Pythagorean triples which solve the quadratic case have been known since ancient times. Euler has definitely proven the cases three and four.

He said he has proof all, but no one knows.

 

[martinbn]

N=3,4,5 has been proved hundreds of years ago, if let them mix and mix in my proof can get N more dense, but far away from complete dense. Only Andrew Wiles has the full proof.

As i said, it is not your proof, it is just an observation that everyone can see.

And if you know you dont get all exponents, why did you ask if you had proven the theorem!

 

[fenstip]

As i said, it is not your proof, it is just an observation that everyone can see.

And if you know you dont get all exponents, why did you ask if you had proven the theorem!

There is no link out there saying human beings except AndrewWiles, improve the number of n to 1000, unless visiting this page, then "everyone" can see it is not my proof including you.

 

[martinbn]

There is no link out there saying human beings except AndrewWiles, improve the number of n to 1000, unless visiting this page, then "everyone" can see it is not my proof including you.

I am not sure what you mean here. Every elementary book on number theory that talks about Fermat says the obvious that of $n=mk$, then the equations becomes $(x^m)^k+(y^m)^k=(z^m)^k$. So it is enough to prove it for $n=4$ and $n$ a prime number.

Also you should look into the history of the problem. There are many people you improved on between Fermat and Wiles.

 

[mfb]

He said he has proof all, but no one knows.

He didn't just say that, he published the proof. Others have checked it and verified that it is correct after fixing a smaller issue in an earlier version.
Edit: I thought this was about Wiles, my comment is about Wiles.

If there is no solution for n then there is no solution for 2n, 3n, 4n, ... - that is very easy to demonstrate, and that's the observation you made here. That alone doesn't prove the theorem, however. You also need to show that there is no solution for all odd primes and you need to show that there is no solution for 4. People had found proofs for 4 and for many primes before, but Wiles was the first one to find a proof for all primes.

 

[fresh_42]

He [Fermat] didn't just say that, he published the proof.

Fermat didn't publish anything. All we have are letters he sent to colleagues, which at the time were basically challenges, and what has been published after his death, was found in his left properties. E.g.

Bernard Frénicle de Bessy published the first proof for the case n=4 as early as 1676. His solution came from Fermat himself, whose proof sketch in this case is known in a marginal note in his Diophant edition on a closely related problem (see Infinite Descent).

Leonhard Euler published a proof for the case n=4 in 1738. Later, with the help of complex numbers, he was also able to confirm the claim for the case n=3, which he published in 1770 (he announced in a letter in 1753 that he had the proof). However, Euler was unable to extend his method of proof to other cases.

At least 20 different pieces of evidence have now been found for the case n=4. For n=3 there are at least 14 different pieces of evidence.

 

[fenstip]

I am not sure what you mean here. Every elementary book on number theory that talks about Fermat says the obvious that of $n=mk$, then the equations becomes $(x^m)^k+(y^m)^k=(z^m)^k$. So it is enough to prove it for $n=4$ and $n$ a prime number.

Also you should look into the history of the problem. There are many people you improved on between Fermat and Wiles.

Even if they/you own the enough knowledge to prove it, it doesn't mean you notice how to prove it. All of our words learning from parents/books/films etc which are full in the proof, so there wouldn't "proof" existed?

 

[fenstip]

He didn't just say that, he published the proof. Others have checked it and verified that it is correct after fixing a smaller issue in an earlier version.

If there is no solution for n then there is no solution for 2n, 3n, 4n, ... - that is very easy to demonstrate, and that's the observation you made here. That alone doesn't prove the theorem, however. You also need to show that there is no solution for all odd primes and you need to show that there is no solution for 4. People had found proofs for 4 and for many primes before, but Wiles was the first one to find a proof for all primes.

I think I know what the difference between us. You are focusing on the proof, I know I referred Fermat's in my observation. I focus on the result (after proving) which I think others ignored 300 years.

 

[fresh_42]

I think I know what the difference between us. You are focusing on the proof, I know I referred Fermat's in my observation. I focus on the result (after proving) which I think others ignored 300 years.

This does not make any sense. Who ignored what? Read my post #13.

Only Fermat's name was given to the problem because of his famous note on the margin of his book. His contributions to any part of the solution were marginal, too.

There is nothing more to discuss on that specific conclusion that was best summarized in @martinbn's post #11. This thread is closed.