On Pillai's problem with the Fibonacci and Pell sequences

Abstract

Let (Fn)n≥0 and (Pn)n≥0 be the Fibonacci and Pell sequences given by the initial conditions F0 = 0, F1 = 1, P0 = 0, P1 = 1 and the recurrence formulas Fn+2 = Fn+1 + Fn, Pn+2 = 2Pn+1 + Pn for all n ≥ 0, respectively. In this note, we study the Pillai type problem: Fn − Pm = Fn1 − Pm1 in non-negative integer pairs (n,m) = (n1,m1). We completely solve this equation.