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We know that $a^n + b^n = c^n$ does not have a solution if $n > 2$ and $a,b,c,n \in \mathbb{N}$, but what if $n \in \mathbb{R}$? Do we have any statement for that?

I was thinking about this but could not find any immediate counter examples.

Specifically, can $a^\pi + b^\pi = c^\pi$ for $a,b,c \in \mathbb{N}$?

I found this. It has a existential proof that $\exists \ n \in \mathbb{R}$ for any $(a,b,c)$

The question remains open for $n = \pi$.

This question is just for fun to see if we can some up with some simple proof :)

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  • $\begingroup$ See wolframalpha.com/input/…, where Wolframalpha does not seem able to find any solutions. $\endgroup$ – Chad Shin Mar 26 '16 at 3:10
  • $\begingroup$ @ChadShin $a,b,c \in \mathbb{N}$ and not $\mathbb{Z}$ $\endgroup$ – Banach Tarski Mar 27 '16 at 9:57
  • $\begingroup$ That doesn't matter. Wolframalpha does not know any non-zero integer solutions. This seems to imply there are no natural solutions. $\endgroup$ – Chad Shin Mar 27 '16 at 14:06
  • $\begingroup$ @Chad Shin Wolfram Alpha doesn't know any non zero solutions to $a^3 + b^3 = c^3$ but that does not put us in any better position to prove Fermat's Last Theorem :) $\endgroup$ – Banach Tarski Mar 27 '16 at 16:31
  • $\begingroup$ Which is why I did not post this as an answer. It seems that it probably isn't. Your problem is very interesting. $\endgroup$ – Chad Shin Mar 27 '16 at 16:48
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The Wikipedia article on Fermat's last theorem has a full section about it, with plenty of references. Here are a few results (see the article for precise references):

  • The equation $a^{1/m} + b^{1/m} = c^{1/m}$ has solutions $a = rs^m$, $b = rt^m$ and $c = r(s+t)^m$ with positive integers $r,s,t>0$ and $s,t$ coprime.
  • When $n > 2$, the equation $a^{n/m} + b^{n/m} = c^{n/m}$ has integer solutions iff $6$ divides $m$.
  • The equation $1/a + 1/b = 1/c$ has solutions $a = mn + m^2$, $b = mn + n^2$, $c = mn$ with $m,n$ positive and coprime integers.
  • For $n = -2$, there are again an infinite number of solutions.
  • For $n < -2$ an integer, there can be no solution, because that would imply that there are solutions for $|n|$.

I don't know if anything is known for irrational exponents.

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