On Pillai’s problem with the Fibonacci and Pell sequences

Abstract

Let $$(F_n)_{n\ge 0}$$ and $$(P_n)_{n\ge 0}$$ be the Fibonacci and Pell sequences given by the initial conditions $$F_0=0,~F_1=1$$ , $$P_0=0, ~P_1=1$$ and the recurrence formulas $$F_{n+2}=F_{n+1}+F_n$$ , $$P_{n+2}=2P_{n+1}+P_n$$ for all $$n\ge 0$$ , respectively. In this note, we study the Pillai type problem: $$\begin{aligned} F_n-P_m=F_{n_1}-P_{m_1} \end{aligned}$$ in non-negative integer pairs $$(n,m)\ne (n_1,m_1)$$ . We completely solve this equation.