Pub Date : 2023-11-21DOI: 10.7546/nntdm.2023.29.4.724-736
Brenda Navarro-Flores, José M. González-Barrios, Raúl Rueda
Prime numbers have been always of great interest. In this work, we explore the prime numbers from a sieve other than the Eratosthenes sieve. Given a prime number $p$, we consider the binary expansion of $frac{1}{p}$ and, in particular, the size of the period of $frac{1}{p}$. We show some results that relate the size of the period to properties of the prime numbers.
{"title":"Binary expansions of prime reciprocals","authors":"Brenda Navarro-Flores, José M. González-Barrios, Raúl Rueda","doi":"10.7546/nntdm.2023.29.4.724-736","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.724-736","url":null,"abstract":"Prime numbers have been always of great interest. In this work, we explore the prime numbers from a sieve other than the Eratosthenes sieve. Given a prime number $p$, we consider the binary expansion of $frac{1}{p}$ and, in particular, the size of the period of $frac{1}{p}$. We show some results that relate the size of the period to properties of the prime numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139252958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.7546/nntdm.2023.29.4.682-694
Ahmet Tekcan, Esra Zeynep Türkmen
In this work, the general terms of almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers of first and second type are determined in terms of balancing and Lucas-balancing numbers. Later some relations on all almost balancing numbers and all almost balancers are obtained. Further the general terms of all balancing numbers, Pell numbers and Pell–Lucas number are determined in terms of almost balancers, almost Lucas-balancers, almost cobalancers and almost Lucas-cobalancers of first and second type.
{"title":"Almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers","authors":"Ahmet Tekcan, Esra Zeynep Türkmen","doi":"10.7546/nntdm.2023.29.4.682-694","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.682-694","url":null,"abstract":"In this work, the general terms of almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers of first and second type are determined in terms of balancing and Lucas-balancing numbers. Later some relations on all almost balancing numbers and all almost balancers are obtained. Further the general terms of all balancing numbers, Pell numbers and Pell–Lucas number are determined in terms of almost balancers, almost Lucas-balancers, almost cobalancers and almost Lucas-cobalancers of first and second type.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"50 38","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.7546/nntdm.2023.29.4.695-704
Neşe Ömür, Sibel Koparal, Laid Elkhiri
In this paper, we define the generalized hyperharmonic numbers of order r, H_{n}^{r}left( sigma right) and get some identities involving these numbers by using Euler’s transform.
{"title":"On sums with generalized harmonic numbers via Euler’s transform","authors":"Neşe Ömür, Sibel Koparal, Laid Elkhiri","doi":"10.7546/nntdm.2023.29.4.695-704","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.695-704","url":null,"abstract":"In this paper, we define the generalized hyperharmonic numbers of order r, H_{n}^{r}left( sigma right) and get some identities involving these numbers by using Euler’s transform.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"51 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.7546/nntdm.2023.29.4.670-681
George Grossman, Aklilu Zeleke, Xinyun Zhu
In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.
本文给出了斐波那契数和卢卡斯数的几个新的恒等式。这是通过求解一类非齐次线性递归关系来实现的。
{"title":"Identities for Fibonacci and Lucas numbers","authors":"George Grossman, Aklilu Zeleke, Xinyun Zhu","doi":"10.7546/nntdm.2023.29.4.670-681","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.670-681","url":null,"abstract":"In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"251 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135321706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we establish some explicit formulas of (q,k)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the q-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.
{"title":"New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications","authors":"Irene Magalhães Craveiro, Elen Viviani Pereira Spreafico, Mustapha Rachidi","doi":"10.7546/nntdm.2023.29.4.647-669","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.647-669","url":null,"abstract":"In this paper we establish some explicit formulas of (q,k)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the q-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136062419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-13DOI: 10.7546/nntdm.2023.29.4.635-646
Elif Tan, Umut Öcal
In this study, we introduce a new class of generalized quaternions whose components are dual-generalized complex Horadam numbers. We investigate some algebraic properties of them.
本文引入了一类新的广义四元数，其组成为双广义复Horadam数。我们研究了它们的一些代数性质。
{"title":"On a generalization of dual-generalized complex Fibonacci quaternions","authors":"Elif Tan, Umut Öcal","doi":"10.7546/nntdm.2023.29.4.635-646","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.4.635-646","url":null,"abstract":"In this study, we introduce a new class of generalized quaternions whose components are dual-generalized complex Horadam numbers. We investigate some algebraic properties of them.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.7546/nntdm.2023.29.3.620-634
K. Lalithambigai, P. Gnanachandra
This paper presents a method of constructing topologies on vertex set of a graph G induced by chromatic partition of vertex set of the graph. It introduces colour lower approximation and colour upper approximation of vertex induced subgraphs and acquaints the open and closed sets of the topology generated by chromatic partition on the vertex set of graphs. It explores some of the properties of colour lower approximation and colour upper approximation of vertex induced subgraphs. It also establishes some new subgraphs based on the colour lower approximation and colour upper approximation and some of their properties have been studied.
{"title":"Topological structures induced by chromatic partitioning of vertex set of graphs","authors":"K. Lalithambigai, P. Gnanachandra","doi":"10.7546/nntdm.2023.29.3.620-634","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.620-634","url":null,"abstract":"This paper presents a method of constructing topologies on vertex set of a graph G induced by chromatic partition of vertex set of the graph. It introduces colour lower approximation and colour upper approximation of vertex induced subgraphs and acquaints the open and closed sets of the topology generated by chromatic partition on the vertex set of graphs. It explores some of the properties of colour lower approximation and colour upper approximation of vertex induced subgraphs. It also establishes some new subgraphs based on the colour lower approximation and colour upper approximation and some of their properties have been studied.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43798035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-17DOI: 10.7546/nntdm.2023.29.3.571-588
Y. Soykan, Nejla Özmen, Inci Okumuş
In this paper, we examine generalized Tridovan sequences and treat in detail two cases called Tridovan sequences and Tridovan–Lucas sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. In addition, we give some identities and matrices related to these sequences.
{"title":"On properties of generalized Tridovan numbers","authors":"Y. Soykan, Nejla Özmen, Inci Okumuş","doi":"10.7546/nntdm.2023.29.3.571-588","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.571-588","url":null,"abstract":"In this paper, we examine generalized Tridovan sequences and treat in detail two cases called Tridovan sequences and Tridovan–Lucas sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. In addition, we give some identities and matrices related to these sequences.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41366919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-02DOI: 10.7546/nntdm.2023.29.3.564-570
Louis Rubin
We consider functions $F:mathbb{Z}_{geq 0}rightarrowmathbb{Z}_{geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}timescdotstimes Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.
我们考虑函数$F:mathbb{Z}_{geq 0}rightarrowmathbb{Z}_{geq 0}$，其中存在一个正整数$n$，使得两个条件成立:$F(p)$对每一个素数$p$除$n$，对于$n$和每一个素数$p$的每一个约数$d$，我们有$d$除$F(p)$, $d$除$F(p mod d)$。根据Khrennikov和Nilsson的方法，我们利用等差数列的素数定理，推导出了这样一个表达式$F$在所有素数$p$上的平均值，并恢复了这两位作者的一个定理作为特例。作为应用，我们计算了定义在循环群的乘积$Z_{f_1(p)}timescdotstimes Z_{f_m(p)}$上的多元幂映射的$r$ -周期点的平均值，其中$f_i(t)$是一个多项式。
{"title":"The average value of a certain number-theoretic function over the primes","authors":"Louis Rubin","doi":"10.7546/nntdm.2023.29.3.564-570","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.564-570","url":null,"abstract":"We consider functions $F:mathbb{Z}_{geq 0}rightarrowmathbb{Z}_{geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}timescdotstimes Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47988025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.7546/nntdm.2023.29.3.598-602
Tomáš Riemel
In this paper, we will focus on the study of a special type of exponential Diophantine equations, including a proof. The main contribution of this article is the mentioned type of equations, which can only be solved by the methods of elementary mathematics.
{"title":"On special exponential Diophantine equations","authors":"Tomáš Riemel","doi":"10.7546/nntdm.2023.29.3.598-602","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.3.598-602","url":null,"abstract":"In this paper, we will focus on the study of a special type of exponential Diophantine equations, including a proof. The main contribution of this article is the mentioned type of equations, which can only be solved by the methods of elementary mathematics.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46303632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：