Suppose, $a,b,c \ge 1$ are integers.
Can the question whether the equation $$x^a+y^b+z^c=n$$ has a solution in integers $x,y,z$ for some particular integer $n$ be undecidable ?
I ask because I read that a slightly complication of the equation occuring in Fermat's last theorem can lead to an undecidable case and I wonder whether the given form already is sufficient to achieve this.