TITLE: Formula for an exponent
QUESTION [0 upvotes]: Is there a formula for $n$ in an equation of the form:
$$a^n+b^n=c$$ 
where $a,b,c \in \mathbb{R}$ and $n \in \mathbb{N}$? Is there any theorem that say something about this kind of a problem?
EDIT: Second version is if  $n \in \mathbb{R}$ instead of  $n \in \mathbb{N}$.

REPLY [0 votes]: For any given $c$ and $n$, the points $(a,b)$ such that $a^n + b^n = c$ form a curve in the $a-b$ plane (of course if $n$ is even you'll need $c > 0$ for this to be nonempty).  The union of these curves for $n \in \mathbb N$ will have Lebesgue measure $0$, and thus miss almost every point of that plane.  So in general the problem can't be solved for $n \in \mathbb N$. 
Here are the curves in the first quadrant for $c=1$ and $n=1$ to $10$.  As $n \to \infty$ the curves approach the polygonal curve $[0,1]$ to $[1,1]$ to $[1,0]$.
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