{"title":"$k$-Pell序列中的Fermat和Mersenne数","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":"{\"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}","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}
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.