I was wondering if it is true that the set of solutions of the equation $$ f(x) = y^k $$ is finite, where $f$ is an irreducible integer polynomial of degree $d \ge 2$ and $y \in \mathbb{Z}[x]$, $k \in \mathbb{N}$ with $k \ge 2$.

I searched in the literature but could not find the answer, and trying to prove it myself has proved unsuccesful.

The MO question Polynomials which always assume perfect power values asks the question for the same equation but assumes that it holds for every integer number, so it isn't quite the solution to current question.

Thanks in advance for any help or counterexample.

I was wondering if it is true that the set of integer solutions of the equation $$ f(x) = y^k $$ is finite, where $f$ is an irreducible integer polynomial of degree $d \ge 2$ and $y \in \mathbb{Z}$, $k \in \mathbb{N}$ with $k \ge 3$.

That is, given an integer irreducible polynomial $f$ of degree greater than two, if the set $\{ (x, y, k) \in \mathbb{Z}^3 | k \ge 3, f(x) = y^k \}$ is finite.

I searched in the literature but could not find the answer, and trying to prove it myself has proved unsuccesful.

The MO question Polynomials which always assume perfect power values asks the question for the same equation but assumes that it holds for every integer number, so it isn't quite the solution to current question.

Thanks in advance for any help or counterexample.

Ok, due to me not paying lot of attention to how i've written it down, the question has been misread a lot, so I've rephrased the question. I hope it is more clear now.