论文标题
关于分辨率的Diophantine方程$ u_n + u_m = x^q $
On the resolution of the Diophantine equation $U_n + U_m = x^q$
论文作者
论文摘要
假设$（u_ {n}）_ {n \ geq 0} $是二进制复发序列，具有主导的根$α$，$α> 1 $，而判别$ d $是无方形的。在本文中，我们研究了diophantine方程$ u_n + u_m = x^q $ in Integers $ n \ geq m \ geq 0 $，$ x \ geq 2 $和$ q \ geq 2 $。首先，我们证明使用对数中的线性表单只有固定的$ x $有限的。其次，我们表明，在{\ em em abc-conjecture}的假设下，$（n，m，x，q）$中只有有限的解决方案。为了证明这一点，我们使用了几种经典结果，例如schmidt子空间定理，这是$ s $ units中线性方程的基本定理，以及Siegel的定理，涉及高细胞方程的解决方案数量的有限。
Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $α$ with $α>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation.
