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Is it known whether for any natural number $n$, I can find (infinitely many?) nontrivial integer tuples $$(x_0,\ldots,x_n)$$ such that $$x_0^n + \cdots + x_{n-1}^n = x_n^n?$$

Obviously this is true for $n = 2$.

Thanks.

  • 0
    Perhaps take a look at http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture - seems even finding one solution for a single value of $n>2$ can be tricky.2012-02-09
  • 0
    Hi testcase - I've seen that, but it seems like the exact opposite of what I'm looking for. Thanks anyway, though.2012-02-09
  • 2
    For $n = 3$ see http://math.fau.edu/richman/cubes.htm .2012-02-09
  • 3
    For "infinitely many", you might want to add "relatively prime"; otherwise it doesn't add to the question since you can scale any nontrivial tuple by infinitely many factors.2012-02-09
  • 1
    Last I knew this was known for $3$, $4$, $5$, $7$, $8$, and open for $6$, $9$, $10$.2012-02-09
  • 1
    Tables of results and conjectures : [A Collection of Algebraic Identities By Tito Piezas III](http://sites.google.com/site/tpiezas/Home/) and the [EulerNet](http://euler.free.fr).2012-02-09

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