What is the Connection Between the ABC Conjecture and Fermat's Last Theorem?

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In summary, the conversation discusses various mathematical concepts such as exponents, the ABC conjecture, and Fermat's last theorem. It is mentioned that a proof of the ABC conjecture would also prove Fermat's last theorem and many other famous conjectures in number theory. Moreover, it is pointed out that p, x, y, a, b, and c are always integers when p is a prime number, but z and c cannot be integers for p > 2. The conversation also touches on the fact that Fermat's last theorem applies to all numbers, not just primes. Finally, the conversation delves into the proof that x, y, and z cannot be integers when p > 2.
  • #1
Russell E. Rierson
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5^1 = 1*0 + 5

5^2 = 2*10 + 5

5^3 = 3*40 + 5

5^p = p*a + 5

x^p = p*a + x



x^p = p*a + x

y^p = p*b + y

z^p = p*c + z



...x^p + y^p = z^p



p*a + x + p*b + y = p*c + z

p*[a + b - c] = z - [x + y]


p = [z - (x + y)]/[a + b - c]



http://www.maa.org/mathland/mathtrek_12_8.html [Broken]


Astonishingly, a proof of the ABC conjecture would provide a way of establishing Fermat's last theorem in less than a page of mathematical reasoning. Indeed, many famous conjectures and theorems in number theory would follow immediately from the ABC conjecture, sometimes in just a few lines.



 
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  • #2
how does this prove that x,y and z can't be integers when p>2? Also a,b and as far as I know can be anything you want.
 
  • #3
FulhamFan3 said:
how does this prove that x,y and z can't be integers when p>2? Also a,b and as far as I know can be anything you want.


p, x, y, a,b,c, are always integers > 0, when p is a prime number. When p is 1, a = 0, b = 0, and c = 0.


z and c cannot be a +integer for p > 2.


[x^p - x]/p = a

[y^p - y]/p = b

[z^p - z]/p = c


for example:


[3^2 - 3]/2 = 3

[3^3 - 3]/3 = 8

[3^5 - 3]/5 = 48

[+integer^p - +integer]/p = another +integer.

It works for all prime numbers.


3^2 + 4^2 = 5^2

2*3 + 3 = 3^2

2*6 + 4 = 4^2



2*3 + 4*3 + 3 + 4 = 6*3 + 7 = 5^2
 
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  • #4
Russel Rierson,

I don't know why this is here and no in the number theory section of the math board.

But as long as it's here, you lost me on your ifrst line: 5^1 = 1*0 + 5
 
  • #5
Russell E. Rierson said:
p, x, y, a,b,c, are always integers > 0, when p is a prime number. When p is 1, a = 0, b = 0, and c = 0.


z and c cannot be a +integer for p > 2.

z and c can be integers when p>2. you also have to prove that statement you can't just say it.

Also fermat's theorem as far as I know isn't just for primes, it's for all numbers. Unless this is some other theorem by fermat. In which case you need to state it so we know what your talking about.
 
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  • #6
FulhamFan3 said:
z and c can be integers when p>2. you also have to prove that statement you can't just say it.

Also fermat's theorem as far as I know isn't just for primes, it's for all numbers. Unless this is some other theorem by fermat. In which case you need to state it so we know what your talking about.


If p = 2 :

x^2 = 2*[1 + 2 + 3+...+ x-1] + x

2^2 = 2*[1] + 2

3^2 = 2*[1 + 2] + 3

4^2 = 2*[1 + 2 + 3] + 4

5^2 = 2*[1 + 2 + 3 + 4] + 5

etc...

This congruence does not hold for p > 2
 
  • #7
5^2 = 2*[1 + 2 + 3 + 4] + 5


5^2 = 2*[1 + 2] + 2*[3 + 4] + 5



5^2 = 2*[1 + 2] + 2*[3 + (4 - 1)] + 5 + 2*1



5^2 = 2*[1 + 2] + 2*[3 + 3] + 7



5^2 = 2*[1 + 2] + 3 + 2*[ 1 + 2 + 3] + 4



5^2 = 3^2 + 4^2
 

1. What is the "Lost Proof of Fermat"?

The "Lost Proof of Fermat" refers to a mathematical proof that was claimed to have been written by the French mathematician Pierre de Fermat in the 17th century. This proof was said to solve Fermat's Last Theorem, which 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.

2. Why is the "Lost Proof of Fermat" important?

The "Lost Proof of Fermat" is important because it could potentially provide a solution to Fermat's Last Theorem, one of the most famous and longstanding problems in mathematics. It could also shed light on Fermat's mathematical genius and his ability to solve complex problems.

3. Has the "Lost Proof of Fermat" ever been found?

No, the "Lost Proof of Fermat" has never been found. It was never published by Fermat during his lifetime and there is no evidence that it ever existed. However, many mathematicians have attempted to recreate the proof based on Fermat's other works and notes.

4. What efforts have been made to find the "Lost Proof of Fermat"?

Many mathematicians have dedicated their careers to studying Fermat's Last Theorem and attempting to find the "Lost Proof of Fermat". Some have searched through Fermat's personal papers and letters for any mention of the proof, while others have tried to recreate it using his other works as a guide. Some have even used advanced computer algorithms to search for patterns and solutions to the theorem.

5. Is it possible that the "Lost Proof of Fermat" will ever be found?

It is unlikely that the "Lost Proof of Fermat" will ever be found, as there is no evidence that it ever existed in the first place. However, the search for the proof has led to many advancements in mathematics and has inspired countless mathematicians to continue studying Fermat's Last Theorem and other unsolved problems. So, even if the proof is never found, its impact on the field of mathematics will continue to be felt.

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