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 of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial such that the set contains a perfect integer power if and only if it belongs to . 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 determined in advance.

中文翻译：

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逆向工程丢番图方程

我们回答了 Samir Siksek 在“Rational Points 2022”会议开放问题会议上提出的问题，从更广泛的意义上来说，这个问题可以被视为丢番图方程的逆向工程。对于任何有限的完美整数幂集，使用 Mihăilescu 定理，我们构造一个多项式，使得该集合包含一个完美整数幂当且仅当它属于 。我们首先讨论更简单的情况，即我们限制具有相同指数的所有幂。在这种情况下，构造的多项式受到龙格方法和费马大定理的启发。因此我们可以构造一个多项式指数丢番图方程，其解是预先确定的。