# Extension of Fermat's theorem?

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

## Answers and Replies

matt grime
Science Advisor
Homework Helper
Every number is the sum of 4 squares. That is one known result.

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

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.

uart
Science Advisor
like, has it been proved, disproved, etc.
Alphanumeric has already posted links that contain counter examples. For example they show several solutions to $a^5 + b^5 + c^5 + d^5 = e^5$

Alphanumeric has already posted links that contain counter examples. For example they show several solutions to $a^5 + b^5 + c^5 + d^5 = e^5$

thanks