Pillai's problem with k-Fibonacci and Pell numbers

Abstract

The k-Fibonacci sequence $\{F_n^{(k)}{\}}_n$ starts with the values $0,\dots ,0,1$ (a total of k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-Fibonacci number and a Pell number. This paper continues and extends the previous work of [J.J. Bravo, C.A. Gómez, and J.L. Herrera, On the intersection of k-Fibonacci and Pell numbers, Bull. Korean Math. Soc. 56(2) (2019), pp. 535–547; S. Hernández, F. Luca, and L.M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Soc. Mat. Mex. 25 (2019), pp. 495–507 and M.O. Hernane, F. Luca, S.E. Rihane, and A. Togbé, On Pillai's problem with Pell numbers and powers of 2, Hardy- Ramanujan J. 41 (2018), pp. 22–31].

Keywords:

• Pillai's problem
• k-Fibonacci number
• Pell number
• linear form in logarithms
• reduction method

2010 Mathematics Subject Classifications:

• 11B39
• 11J86

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A linear recurrence sequence has a dominant root if its characteristic polynomial has a simple root α whose absolute value exceeds the absolute values of the remaining roots.

Additional information

Funding

J. J. B. was supported in part by Project VRI ID 5385 (Universidad del Cauca). M. D. thanks to the Universidad del Cauca for its support during her doctoral studies. C. A. G. was supported in part by Project 71280 (Universidad del Valle).