Simultaneous powers Let $a,b,c \in \mathbb{Z}$ be coprime. Is is possible to have
\begin{align*}
a+b = x^n \\
b+c = y^n \\
c+a = z^n
\end{align*}
for $x,y,z \in \mathbb{Z}$ and $n$ a positive integer? If it helps, assume $n$ is an odd prime.

Yes it is possible. For instance $199, 530,$ and $801$ are pairwise coprime, and any two of them sum to a cube.
A smaller example is $36, 307,$ and $693$, which have no common factors for all three; but are not pairwise coprime.
Both examples were found with a computer search.
