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On the intersections of Fibonacci, Pell, and Lucas numbers. (English) Zbl 1228.11018

The author describes 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. He proves 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. The intersection in these cases also forms a Lucas sequence. In particular 0, 1, 2, and 5 are the only numbers that are both Fibonacci and Pell. Similar results are listed for other pairs of Lucas sequences. Lucas sequences with arbitrary initial terms are discussed.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11D72 Diophantine equations in many variables
11Y50 Computer solution of Diophantine equations
11Y55 Calculation of integer sequences

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