On the Exponential Diophantine Equation $(F_{m+1}^{(k)})^x-(F_{m-1}^{(k)})^x = F_n^{(k)}$

Hayat Bensella; Bijan Kumar Patel; Djilali Behloul

paper

On the Exponential Diophantine Equation $(F_{m + 1}^{(k)})^{x} - (F_{m - 1}^{(k)})^{x} = F_{n}^{(k)}$

arXiv:2205.13168

Abstract

In this paper, we explicitly find all solutions of the title Diophantine equation, using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker-Davenport reduction method in Diophantine approximation, due to Dujella and Pethö. This paper extends the previous work of \cite{Patel}.

On the Exponential Diophantine Equation (Fm+1(k)​)x−(Fm−1(k)​)x=Fn(k)​

Abstract

On the Exponential Diophantine Equation $(F_{m + 1}^{(k)})^{x} - (F_{m - 1}^{(k)})^{x} = F_{n}^{(k)}$