On the intersections of Fibonacci, Pell, and Lucas numbers

TL;DR

This paper investigates the intersections of various Lucas sequences, including Fibonacci, Pell, and Lucas numbers, proving finiteness of intersections except in specific cases, and solving related Diophantine equations.

Contribution

It characterizes the intersections of Lucas sequences, identifying when they are finite or infinite, and extends results to sequences with arbitrary initial terms.

Findings

Only 0, 1, 2, and 5 are both Fibonacci and Pell numbers.

Intersections are finite except in specific cases involving discriminants.

The intersection sequences, when infinite, also form Lucas sequences.

Abstract

We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the product of their discriminants is a perfect square. Moreover, the intersection in these cases also forms a Lucas sequence. Our approach relies on solving homogeneous quadratic Diophantine equations and Thue equations. In particular, we prove that 0, 1, 2, and 5 are the only numbers that are both Fibonacci and Pell, and list similar results for many other pairs of Lucas sequences. We further extend our results to Lucas sequences with arbitrary initial terms.