Fibonacci Numbers that Are $$\eta$$-concatenations of Leonardo and Lucas Numbers

Abstract

Let $$\{F_{r}\}_{r\geq0}$$, $$\{L_{r}\}_{r\geq0}$$ and $$\{Le_{r}\}_{r\geq0}$$ be $$r$$-th terms of Fibonacci, Lucas and Leonardo sequences, respectively. In this paper, we determined the effective bounds for the solutions of the Diophantine equation $$F_{r}=\eta^{k}Le_{s}+L_{t}$$ in non-negative integers $$r$$, $$s$$, $$t$$, where $$k$$ represents the number of digits of $$L_{t}$$ in base $$\eta\geq2$$. In addition, we applied linear forms in logarithms of algebraic numbers and the reduction method based on the continued fraction. In particular, we investigated all solutions of this Diophantine equation for $$\eta\in[2,10]$$.