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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. 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Alekseyev","submitted_at":"2010-02-08T20:08:08Z","abstract_excerpt":"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. 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