{"title":"Fermat and Mersenne numbers in $k$-Pell sequence","authors":"B. Normenyo, S. Rihane, A. Togbé","doi":"10.30970/ms.56.2.115-123","DOIUrl":null,"url":null,"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.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.56.2.115-123","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
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.