On Pillai's problem with the Fibonacci and Pell sequences
Let \( (F_n)_{n\ge 0} \) and \( (P_n)_{n\ge 0} \) be the sequences of Fibonacci and Pell numbers given by \( F_0=P_0=0, ~ F_1 = P_1=1 \), \( F_{n+2}=F_{n+1}+F_n \), and \( P_{n+2}=2P_{n+1}+P_n \) for all \( n\ge 0 \), respectively. In the paper under review, the authors study the Pillai type problem: \begin{align*} F_n-P_m = F_{n_1}-P_{m_1} = c\tag{1} \end{align*} in non-negative integer pairs \( (n,m)\neq (n_1, m_1) \). They completely solve the Diophantine equation (1). In their main result, they prove that the only integers \( c \) admitting representations as a difference between a Fibonacci number and a Pell number in at least two different ways belong to the set: \begin{align*} c\in\{ -4,-2, -1, 0, 1, 2, 3, 5, 8, 19\}.\tag{2} \end{align*} Furthermore, they explicitly list down the representations for each \(c\) in (2) in the form (1). To prove the main result, the authors use the elementary properties of the Fibonacci and Pell sequences, the Baker's theory for linear forms in logarithms of algebraic numbers to effectively bound their variables, and the Baker-Davenport reduction procedure to reduce the bounds to computable sizes. Computations are done with help of a computer program in Mathematica.
scientific article; zbMATH DE number 7137966
scientific article