Reverse engineered Diophantine equations

We answer a question of Samir Siksek, asked at the open problems session of the conference ``Rational Points 2022'', which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set $S$ of perfect integer powers, using Mih\u{a}ilescu's theorem, we construct a polynomial $f_S\in \Z[x]$ such that the set $f_S(\Z)$ contains a perfect integer power if and only if it belongs to $S$. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge's method and Fermat's Last Theorem. Therefore we can construct a polynomial-exponential Diophantine equation whose solutions are described in advance.