Extension of Fermat's theorem?

  • Thread starter ACG
  • Start date
  • #1
ACG
46
0

Main Question or Discussion Point

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

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
Every number is the sum of 4 squares. That is one known result.
 
  • #3
ACG
46
0
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
 
  • #4
289
0
Last edited by a moderator:
  • #5
2
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
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
uart
Science Advisor
2,776
9
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
2
0
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
 

Related Threads on Extension of Fermat's theorem?

  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
3
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
6
Views
2K
  • Last Post
3
Replies
53
Views
14K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
2
Views
2K
Top