70 research outputs found
On the intersections of Fibonacci, Pell, and Lucas numbers
We describe how to compute the intersection of two Lucas sequences of the
forms or
with that includes sequences of Fibonacci, Pell, Lucas, and
Lucas-Pell numbers. We prove that such an intersection is finite except for the
case and and the case of two -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
Fibonacci or Lucas numbers that are products of two Lucas numbers or two Fibonacci numbers
This contribution presents all possible solutions to the Diophantine
equations and . To be clear, Fibonacci numbers that
are the product of two arbitrary Lucas numbers and Lucas numbers that are the
product of two arbitrary Fibonacci numbers are determined herein. The results
under consideration are proven by using Dujella-Peth\"o lemma in coordination
with Matveev's theorem. All common terms of the Fibonacci and Lucas numbers are
determined. Further, the Lucas-square Fibonacci and Fibonacci-square Lucas
numbers are given
A General Approach to Proving Properties of Fibonacci Representations via Automata Theory
We provide a method, based on automata theory, to mechanically prove the
correctness of many numeration systems based on Fibonacci numbers. With it,
long case-based and induction-based proofs of correctness can be replaced by
simply constructing a regular expression (or finite automaton) specifying the
rules for valid representations, followed by a short computation. Examples of
the systems that can be handled using our technique include Brown's lazy
representation (1965), the far-difference representation developed by Alpert
(2009), and three representations proposed by Hajnal (2023). We also provide
three additional systems and prove their validity.Comment: In Proceedings AFL 2023, arXiv:2309.0112
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