On the Intersections of Fibonacci, Pell, and Lucas Numbers
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.
© de Gruyter 2011
Artikel in diesem Heft
- Preface
- On the Intersections of Fibonacci, Pell, and Lucas Numbers
- Proving Balanced T2 and Q2 Identities Using Modular Forms
- Sum-Product Inequalities with Perturbation
- Power Totients with Almost Primes
- On the Sum of Reciprocals of Amicable Numbers
- Tromping Games: Tiling with Trominoes
- There are No Multiply-Perfect Fibonacci Numbers
- Large Zero-Free Subsets of ℤ/pℤ
Artikel in diesem Heft
- Preface
- On the Intersections of Fibonacci, Pell, and Lucas Numbers
- Proving Balanced T2 and Q2 Identities Using Modular Forms
- Sum-Product Inequalities with Perturbation
- Power Totients with Almost Primes
- On the Sum of Reciprocals of Amicable Numbers
- Tromping Games: Tiling with Trominoes
- There are No Multiply-Perfect Fibonacci Numbers
- Large Zero-Free Subsets of ℤ/pℤ
