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