On the resolution of the Diophantine equation $U_n+U_m=x^q$

Abstract

Suppose that $(U_n{)}_{n\ge 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha >1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n+U_m=x^q$ in integers $n\ge m\ge 0$, $x\ge 2$, and $q\ge 2$. 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 $q,x\ge 2$ 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.

20 pages

Cited by in corpus (1)

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