Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Extension of Fermat's theorem?

  1. Mar 25, 2007 #1


    User Avatar

    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,

  2. jcsd
  3. Mar 25, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Every number is the sum of 4 squares. That is one known result.
  4. Mar 25, 2007 #3


    User Avatar


    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.

  5. Mar 28, 2007 #4
  6. Jun 1, 2008 #5
    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.)?
  7. Jun 1, 2008 #6


    User Avatar
    Science Advisor

    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]
  8. Jun 1, 2008 #7
  9. Jun 1, 2008 #8
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Extension of Fermat's theorem?
  1. Fermat's last theorem? (Replies: 2)

  2. Fermat's Last Theorem (Replies: 3)