On Fermat’s last theorem and others....

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In summary, Fermat's last theorem states that for any n except 2, the equation X^n+Y^n=Z^n is not true for any positive integer triplet X, Y and Z. Despite Fermat's claim of having a proof for this theorem, it has remained unsolved for centuries. Many mathematicians have attempted to prove it, but it was eventually proved by Andrew Wiles in 1994. However, Fermat's proof has never been found. The fascination with this theorem stems from Fermat's famous quote "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." But as it turns out, his proof was flawed. Some mathematicians have tried to understand this theorem better
  • #1
PengKuan
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On Fermat’s last theorem

This theorem states that for any n except 2, the equation X^n+Y^n=Z^n is not true for any positive integer triplet X, Y and Z. Fermat’s “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” has fascinated mathematicians from 1637 but no one has found what his proof was. Let us try to understand this theorem better.

Please read the article at
On Fermat’s last theorem
http://pengkuanonmaths.blogspot.com/2015/07/on-fermats-last-theorem.html
or On Fermat’s last theorem
https://www.academia.edu/13665056/On_Fermat_s_last_theorem
 
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  • #3
As expected, the proof is flawed. I don't get the fascination with a very obscure mathematical result that already has a nice proof. Anyway, since

[tex]\prod_{j=1}^i\frac{1/n - j +1}{j} X^n[/tex] is not a natural number in ##\{0,1,...,Y^n\}##, it is not the ##i##'th digit in the expansion in the basis ##Y^n##. So it being repeating or not has no impact whatsoever.

The flaw in the continued fraction thing is clear too since it doesn't satisfy the irrationality criterion.

Furthermore, in neither of your proofs have you really used that ##n\neq 2##.
 
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  • #4
micromass said:
As expected, the proof is flawed. I don't get the fascination with a very obscure mathematical result that already has a nice proof. Anyway, since

[tex]\prod_{j=1}^i\frac{1/n - j +1}{j} X^n[/tex] is not a natural number in ##\{0,1,...,Y^n\}##, it is not the ##i##'th digit in the expansion in the basis ##Y^n##. So it being repeating or not has no impact whatsoever.

The flaw in the continued fraction thing is clear too since it doesn't satisfy the irrationality criterion.

Furthermore, in neither of your proofs have you really used that ##n\neq 2##.

Thanks.

I know this. This is why I asked for help at the end of the article.
 
  • #5
PengKuan said:
On Fermat’s last theorem

This theorem states that for any n except 2, the equation X^n+Y^n=Z^n is not true for any positive integer triplet X, Y and Z. Fermat’s “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” has fascinated mathematicians from 1637 but no one has found what his proof was.
Years after writing this, Fermat published separate proofs for the cases 3 and 4. He would not have done that if he had a proof for "all n". What happened was what happens to everyone- he thought he saw a proof that would work for all n but on later consideration saw that there was an error. Fermat clearly did not have a valid proof.
 
  • #6
HallsofIvy said:
Years after writing this, Fermat published separate proofs for the cases 3 and 4. He would not have done that if he had a proof for "all n". What happened was what happens to everyone- he thought he saw a proof that would work for all n but on later consideration saw that there was an error. Fermat clearly did not have a valid proof.

It appears all his remarks in margin are correct according the youtube video on Numberphile.
 
  • #7
micromass said:
As expected, the proof is flawed. I don't get the fascination with a very obscure mathematical result that already has a nice proof. Anyway, since

[tex]\prod_{j=1}^i\frac{1/n - j +1}{j} X^n[/tex] is not a natural number in ##\{0,1,...,Y^n\}##, it is not the ##i##'th digit in the expansion in the basis ##Y^n##. So it being repeating or not has no impact whatsoever.

The flaw in the continued fraction thing is clear too since it doesn't satisfy the irrationality criterion.

Furthermore, in neither of your proofs have you really used that ##n\neq 2##.

I have changed my proof for continued fraction. It fits the criterion now.

I have added the mention n>2.
 
  • #8
Again, where does your proof fail for ##n=2##?
 
  • #9
micromass said:
Again, where does your proof fail for ##n=2##?

I have computed for X=3 and y=4. The continued fraction converges to 5.

I think for n=2, the Pythagorean triples are exception for the theorem "if a1, a2,… and b1, b2,…are positive integers with ak<bk for all sufficiently large k, then the fraction converges to an irrational limit "

Indeed, all other x and y for n=2 give irrational z.
 
  • #10
PengKuan said:
I think for n=2, the Pythagorean triples are exception for the theorem "if a1, a2,… and b1, b2,…are positive integers with ak<bk for all sufficiently large k, then the fraction converges to an irrational limit "

No, that theorem has no exceptions, since it is definitely proven to be true. So either your proof must somehow not work for ##n=2##, or it is flawed in general.
 
  • #11
micromass said:
No, that theorem has no exceptions, since it is definitely proven to be true. So either your proof must somehow not work for ##n=2##, or it is flawed in general.

You may be right. But I'm unable to find the flaw.

This theorem is about number. All coefficients are numbers. But in my expression the fractions are functions. The pythagorean triple case may be reducible if they were number.

I have just checked my derivation. The coefficients for even k do not fit. So, this is the cause for n=2 case fail. Maybe for n>2 there can be new theorem that proves the irrationality. But I cannot find.
 
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  • #12
PengKuan said:
It appears all his remarks in margin are correct according the youtube video on Numberphile.
What? The NumberPhile video says the exact opposite of what you seem to think it does. On the video about Fermat on "Numberphile" at 4:35, the person clearly says "Fermat thought he had a proof but was mistaken."The remark about Fermat's "remarks in the margin", at 3:15, is that "In every case where Fermat said he had a proof he was correct except this one."

That is essentially what I said.
 
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  • #13
HallsofIvy said:
What? The NumberPhile video says the exact opposite of what you seem to think it does. On the video about Fermat on "Numberphile" at 4:35, the person clearly says "Fermat thought he had a proof but was mistaken."The remark about Fermat's "remarks in the margin", at 3:15, is that "In every case where Fermat said he had a proof he was correct except this one."

That is essentially what I said.

I have mixed up two video that I have seen. Sorry.
 
  • #14
micromass said:
No, that theorem has no exceptions, since it is definitely proven to be true. So either your proof must somehow not work for ##n=2##, or it is flawed in general.
I have just checked my derivation. The coefficients for even k do not fit. So, this is the cause for n=2 case fail. Maybe for n>2 there can be new theorem that proves the irrationality. But I cannot find.
 

FAQ: On Fermat’s last theorem and others....

1. What is Fermat's last theorem?

Fermat's last theorem is a mathematical conjecture proposed by Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

2. Has Fermat's last theorem been proven?

Yes, Fermat's last theorem was proven in 1994 by Andrew Wiles after over 350 years of attempts by mathematicians to prove it. Wiles used techniques from number theory and algebraic geometry to prove the theorem.

3. Are there other theorems similar to Fermat's last theorem?

Yes, there are many other theorems that fall into the category of "unsolved problems." Some notable examples include Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture.

4. Why is Fermat's last theorem important?

Fermat's last theorem is important because it was one of the longest-standing unsolved problems in mathematics, and its proof required the development of new mathematical techniques and theories. It also has many applications in other areas of mathematics, such as number theory and algebraic geometry.

5. Can you explain the significance of Wiles' proof of Fermat's last theorem?

Wiles' proof of Fermat's last theorem is considered one of the greatest achievements in mathematics. It not only solved a centuries-old problem, but also advanced the field of mathematics with the new techniques and theories developed in the process. It also solidified the connections between seemingly disparate areas of mathematics, such as number theory and algebraic geometry.

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