# Can the question whether $x^a+y^b+z^c=n$ has a solution over the integers be undecidable?

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.

• If x, y, z are all positive, then there seems to be only a finite number of possible trials fo r the expression. – herb steinberg Apr 23 '18 at 21:08
• Of course, negative integers $x,y,z$ are allowed as well , otherwise the problem would be trivially decidable. – Peter Apr 23 '18 at 22:42