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Integers


Mathematical Citation Quotient (MCQ) 2017: 0.20

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1867-0660
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On the Intersections of Fibonacci, Pell, and Lucas Numbers

Max A. Alekseyev
Published Online: 2011-06-04 | DOI: https://doi.org/10.1515/integ.2011.021

Abstract

We describe how to compute the intersection of two Lucas sequences of the forms or with P ∈ ℤ that includes sequences of Fibonacci, Pell, Lucas, and Lucas–Pell numbers. We prove that such an intersection is finite except for the case Un (1, –1) and Un (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.

Keywords.: Pell Equations; Thue Equations; Fibonacci Numbers; Lucas Sequences

About the article

Received: 2010-02-08

Revised: 2010-11-05

Accepted: 2011-01-24

Published Online: 2011-06-04

Published in Print: 2011-06-01


Citation Information: Integers, Volume 11, Issue 3, Pages 239–259, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integ.2011.021.

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