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Extension of Fermat's theorem?

  1. Mar 25, 2007 #1


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    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

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


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    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
    Last edited by a moderator: Apr 22, 2017
  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


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    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
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