Alekseyev, Max A. On the intersections of Fibonacci, Pell, and Lucas numbers. (English) Zbl 1228.11018 Integers 11, No. 3, 239-259, A1 (2011). 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. Reviewer: Florin Nicolae (Berlin) Cited in 2 ReviewsCited in 15 Documents 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 Keywords:Fibonacci numbers; Lucas sequences; Pell equations; Thue equations Software:PARI/GP PDFBibTeX XMLCite \textit{M. A. Alekseyev}, Integers 11, No. 3, 239--259, A1 (2011; Zbl 1228.11018) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Integers m that belong to at least two distinct Lucas sequences U(P,Q) with P>0 different from m, |Q|=1, and (P,Q) different from (3,1). Integers m that belong to Lucas sequences U(P1,Q1) and V(P2,Q2) with P1, P2 > 0 both different from m, |Q1|=|Q2|=1.