Extension of Fermat's theorem?

In summary, the conversation is about an extension to Fermat's last theorem where the question is posed for which values of k does the equation x1^n + x2^n + ... +xk^n = z^n have solutions for n > k. Alphanumeric provides links to counterexamples and mentions that it is hard to give specifics as many cases are still unproven. The conversation also briefly touches on the sum of four squares, with a link provided for more information.
  • #1
ACG
46
0
Hi! I assume you all know Fermat's last theorem. Well, has anyone considered the following extension to it? Assuming we're just using integers:

We know that x1^n + x2^n = y^n has no solution for n > 2. However, what about this?

For which values of k does x1^n+x2^n+...+xk^n = z^n have solutions for n > k?

We know it's true for 1 ((-1)^2 = 1^2) and not true for 2. What about other numbers?

Thanks in advance,

ACG
 
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  • #2
Every number is the sum of 4 squares. That is one known result.
 
  • #3
Hi!

That's cool -- I'd wondered about that myself :)

However, that won't help here: the sum of four objects case would have to be a^5+b^5+c^5+d^5 = e^5.

ACG
 
  • #5
ACG said:
Hi! I assume you all know Fermat's last theorem. Well, has anyone considered the following extension to it? Assuming we're just using integers:

We know that x1^n + x2^n = y^n has no solution for n > 2. However, what about this?

For which values of k does x1^n+x2^n+...+xk^n = z^n have solutions for n > k?

We know it's true for 1 ((-1)^2 = 1^2) and not true for 2. What about other numbers?

Thanks in advance,

ACG

Yes, I have (considered that, and also wrote a short program to try and find a counter-example).
Have you been able to find anymore details about this one (like, has it been proved, disproved, etc.)?
Thanks.
 
  • #6
like, has it been proved, disproved, etc.
Alphanumeric has already posted links that contain counter examples. For example they show several solutions to [itex]a^5 + b^5 + c^5 + d^5 = e^5[/itex]
 
  • #7
uart said:
Alphanumeric has already posted links that contain counter examples. For example they show several solutions to [itex]a^5 + b^5 + c^5 + d^5 = e^5[/itex]

thanks
 

1. What is Fermat's theorem?

Fermat's theorem, also known as Fermat's last theorem, is a mathematical theorem that 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. What is an extension of Fermat's theorem?

An extension of Fermat's theorem is a variation or a generalization of the original theorem, which may have different conditions or constraints, but is based on the same principles and ideas as the original theorem.

3. Why is the extension of Fermat's theorem important?

The extension of Fermat's theorem is important because it helps to further our understanding of number theory and the relationships between different mathematical concepts. It also allows us to apply similar principles to different problems and situations.

4. What are some examples of extensions of Fermat's theorem?

There are many examples of extensions of Fermat's theorem, such as the generalization to higher dimensions, the introduction of complex numbers, and the inclusion of additional variables in the equation. Other examples include the extension to polynomials and the application to other areas of mathematics such as geometry and topology.

5. What are the implications of the extension of Fermat's theorem?

The extension of Fermat's theorem has significant implications in mathematics, as it helps to uncover new insights and connections between different areas of study. It also has practical applications in fields such as cryptography and coding theory, where it is used to develop secure algorithms and protocols.

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