On Pell Numbers Representable as a Product of Two Generalized Fibonacci Numbers
Abstract
A generalization of the well-known Fibonacci sequence is the $ k $ -Fibonacci sequence for a fixed integer $ k\geq 2 $ . The first $ k $ terms of this sequence are $ 0,0,\dots,1 $ , and each subsequent term is the sum of the preceding $ k $ terms. In this paper, we find all Pell numbers that can be written as a product of two $ k $ -Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a variation of a result of Dujella and Pethő in Diophantine approximation. This work generalizes a prior result of Alekseyev on the intersection of the Fibonacci and Pell sequences, a result of Ddamulira, Luca, and Rakotomalala on Pell numbers that are products of two Fibonacci numbers, and a result of Bravo, Gómez, and Herrera on Pell numbers that appear in the $ k $ -Fibonacci sequence.
Fields of Study & Subjects
Advanced Mathematical Theories and Applications
Advanced Combinatorial Mathematics
semigroups and automata theory
Publication details
- Journal / source
- Siberian Mathematical Journal
- Publisher
- Pleiades Publishing Ltd
- Volume
- 67
- Issue
- 4
- Pages
- 896–905
- Publication date
- 1 July 2026
- Language
- EN