Integral solutions $(a,b,c)$ for $a^\pi + b^\pi = c^\pi$

Question

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 :)

Answer 1

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.