# Fermat and Mersenne numbers in $k$-Pell sequence

• B. Normenyo Department of Mathematics, University of Ghana
• S. Rihane Department of Mathematics and Computer Sciences University Center Abdelhafid Boussouf Mila, Algeria
• A. Togbe Department of Mathematics and Statistics Purdue University Northwest Westville, USA
Keywords: k-Pell number, Fermat number, Mersenne number, Fibonacci number, Linear form in logarithms, Reduction algorithm

### Abstract

For an integer $k\geq 2$, let $(P_n^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence, which starts with $0,\ldots,0,1$ ($k$ terms) and each term afterwards is defined by the recurrence
$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)},\quad \text{for all }n \geq 2.$
For any positive integer $n$, a number of the form $2^n+1$ is referred to as a Fermat number, while a number of the form $2^n-1$ is referred to as a Mersenne number. The goal of this paper is to determine Fermat and Mersenne numbers which are members of the $k$-generalized Pell sequence. More precisely, we solve the Diophantine equation $P^{(k)}_n=2^a\pm 1$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$. We prove a theorem which asserts that, if the Diophantine equation $P^{(k)}_n=2^a\pm 1$ has a solution $(n,a,k)$ in positive integers $n, k, a$ with $k \geq 2$, $a\geq 1$, then we must have that $(n,a,k)\in \{(1,1,k),(3,2,k),(5,5,3)\}$. As a result of our theorem, we deduce that the number $1$ is the only Mersenne number and the number $5$ is the only Fermat number in the $k$-Pell sequence.

### Author Biographies

B. Normenyo, Department of Mathematics, University of Ghana

Department of Mathematics, University of Ghana

S. Rihane, Department of Mathematics and Computer Sciences University Center Abdelhafid Boussouf Mila, Algeria

Department of Mathematics and Computer Sciences
University Center Abdelhafid Boussouf
Mila, Algeria

A. Togbe, Department of Mathematics and Statistics Purdue University Northwest Westville, USA

Department of Mathematics and Statistics
Purdue University Northwest
Westville, USA

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Published
2021-12-26
How to Cite
Normenyo, B., Rihane, S., & Togbe, A. (2021). Fermat and Mersenne numbers in $k$-Pell sequence. Matematychni Studii, 56(2), 115-123. https://doi.org/10.30970/ms.56.2.115-123
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