2025/07/18 by Bravo, Jhon J., Das, Pranabesh, Herrera, Jose L. +1
#11B39 #11J86 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2507.13674
A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequence with some fixed integer k≥ 2. The first k terms of this sequence are 0,0, …, 1, and each term afterwards 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 which dealt with determining the intersection of the Fibonacci and Pell sequences, a work of Ddamulira, Luca and Rakotomalala who searched for Pell numbers which are products of two Fibonacci numbers, and a result of Bravo, Gómez, and Herrera, who found all Pell numbers appearing in the k-Fibonacci sequence.