论文标题
反设计的二聚体方程
Reverse engineered Diophantine equations
论文作者
论文摘要
我们回答了Samir Siksek的一个问题，在会议的公开问题会议上询问``Rational Points 2022''的问题，从广义上讲，它可以看作是Diophantine方程的反向工程。对于任何有限的完美整数功率的$ s $，使用Mihăilescu定理，我们在\ z [x] $中构建一个多项式$ f_s \，使得set $ f_s（\ z）$在且仅在属于$ s $的情况下包含完美的整数功率。我们首先讨论一个更轻松的情况，我们将其限制在具有相同指数的所有权力上。在这种情况下，构造的多项式受到Runge的方法和Fermat的最后定理的启发。因此，我们可以构建一个多项式指数的二一个方程，其溶液被事先描述。
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 Mihăilescu'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.
