On the intersections of Fibonacci, Pell, and Lucas numbers

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.

💡 Research Summary

The paper investigates the intersection of two Lucas sequences of the forms (U_n(P,\pm1)) and (V_n(P,\pm1)) with integer parameter (P). These families encompass the classic Fibonacci, Pell, Lucas, and Lucas‑Pell sequences, as well as many other linear recurrences with characteristic equation (x^2-Px\pm1=0). The central question is whether the set of integers that belong to both sequences is finite or infinite, and, in the infinite case, what algebraic structure the intersection possesses.

The authors begin by recalling the closed‑form expressions for the two families:
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