  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.
