Pub Date : 2023-05-16DOI: 10.7546/nntdm.2023.29.2.360-371
M. Rana, H. Kaur, K. Garg
In this paper, we provide some recurrence relations connecting restricted partition functions and mock theta functions. Elementary manipulations are used including Jacobi triple product identity, Euler’s pentagonal number theorem, and Ramanujan’s theta functions for finding the recurrence relations.
{"title":"Recurrence relations connecting mock theta functions and restricted partition functions","authors":"M. Rana, H. Kaur, K. Garg","doi":"10.7546/nntdm.2023.29.2.360-371","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.360-371","url":null,"abstract":"In this paper, we provide some recurrence relations connecting restricted partition functions and mock theta functions. Elementary manipulations are used including Jacobi triple product identity, Euler’s pentagonal number theorem, and Ramanujan’s theta functions for finding the recurrence relations.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42051893","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}
{"title":"On the multiplicative group generated by Big{{[sqrt {2}n]over n}~mid~ninmathbb{N} Big}. V","authors":"I. Kátai, B. M. Phong","doi":"10.7546/nntdm.2023.29.2.348-353","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.348-353","url":null,"abstract":"Let $f,g$ be completely multiplicative functions, $vert f(n)vert=vert g(n)vert =1 (ninmathbb{N})$. Assume that $${1over {log x}}sum_{nle x}{vert g([sqrt{2}n])-Cf(n)vertover n}to 0 quad (xtoinfty).$$ Then $$f(n)=g(n)=n^{itau},quad C=(sqrt{2})^{itau}, tauin mathbb{R}.$$","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135337321","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 : 2023-05-11DOI: 10.7546/nntdm.2023.29.2.336-347
B. Kuloğlu, E. Özkan, A. Shannon
In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called ‘incomplete’ number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.
{"title":"p-Analogue of biperiodic Pell and Pell–Lucas polynomials","authors":"B. Kuloğlu, E. Özkan, A. Shannon","doi":"10.7546/nntdm.2023.29.2.336-347","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.336-347","url":null,"abstract":"In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called ‘incomplete’ number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48582017","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 : 2023-05-09DOI: 10.7546/nntdm.2023.29.2.322-335
Munesh Kumari, K. Prasad, R. Frontczak
In this paper, we introduce a new family of sequences called the k-Fibonacci and k-Lucas spinors. Starting with the Binet formulas we present their basic properties, such as Cassini’s identity, Catalan’s identity, d’Ocagne’s identity, Vajda’s identity, and Honsberger’s identity. In addition, we discuss their generating functions. Finally, we obtain sum formulae and relations between k-Fibonacci and k-Lucas spinors.
{"title":"On the k-Fibonacci and k-Lucas spinors","authors":"Munesh Kumari, K. Prasad, R. Frontczak","doi":"10.7546/nntdm.2023.29.2.322-335","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.322-335","url":null,"abstract":"In this paper, we introduce a new family of sequences called the k-Fibonacci and k-Lucas spinors. Starting with the Binet formulas we present their basic properties, such as Cassini’s identity, Catalan’s identity, d’Ocagne’s identity, Vajda’s identity, and Honsberger’s identity. In addition, we discuss their generating functions. Finally, we obtain sum formulae and relations between k-Fibonacci and k-Lucas spinors.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45342390","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 : 2023-05-09DOI: 10.7546/nntdm.2023.29.2.310-321
Gonca Kızılaslan, Leyla Karabulut
We define a generalization of Tribonacci and Tribonacci–Lucas quaternions with arbitrary Tribonacci numbers and Tribonacci–Lucas numbers coefficients, respectively. We get generating functions and Binet’s formulas for these quaternions. Furthermore, several sum formulas and a matrix representation are obtained.
{"title":"Unrestricted Tribonacci and Tribonacci–Lucas quaternions","authors":"Gonca Kızılaslan, Leyla Karabulut","doi":"10.7546/nntdm.2023.29.2.310-321","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.310-321","url":null,"abstract":"We define a generalization of Tribonacci and Tribonacci–Lucas quaternions with arbitrary Tribonacci numbers and Tribonacci–Lucas numbers coefficients, respectively. We get generating functions and Binet’s formulas for these quaternions. Furthermore, several sum formulas and a matrix representation are obtained.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45234967","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 : 2023-05-02DOI: 10.7546/nntdm.2023.29.2.284-309
Zahra Amroune, D. Bellaouar, Abdelmadjid Boudaoud
For any positive integer $n$ let $dleft( nright) $ and $varphi left( nright) $ be the number of divisors of $n$ and the Euler's phi function of $n$, respectively. In this paper we present some notes on the equation $dleft( n^{2}right) =dleft( varphi left( nright) right).$ In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson's conjecture implies that $dleft( n^{2}right) =dleft( varphi left( nright) right) $ infinitely often.
{"title":"A class of solutions of the equation d(n2) = d(φ(n))","authors":"Zahra Amroune, D. Bellaouar, Abdelmadjid Boudaoud","doi":"10.7546/nntdm.2023.29.2.284-309","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.284-309","url":null,"abstract":"For any positive integer $n$ let $dleft( nright) $ and $varphi left( nright) $ be the number of divisors of $n$ and the Euler's phi function of $n$, respectively. In this paper we present some notes on the equation $dleft( n^{2}right) =dleft( varphi left( nright) right).$ In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson's conjecture implies that $dleft( n^{2}right) =dleft( varphi left( nright) right) $ infinitely often.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41760036","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 : 2023-05-02DOI: 10.7546/nntdm.2023.29.2.276-283
Hamilton Brito da Silva
The divisors of a natural number are very important for several areas of mathematics, representing a promising field in number theory. This work sought to analyze new relations involving the divisors of natural numbers, extending them to prime numbers. These are relations that may have an interesting application for counting the number of divisors of any natural number and understanding the behavior of prime numbers. They are not a primality test, but they can be a possible tool for this and could also be useful for understanding the Riemann’s zeta function that is strongly linked to the distribution of prime numbers.
{"title":"New properties of divisors of natural number","authors":"Hamilton Brito da Silva","doi":"10.7546/nntdm.2023.29.2.276-283","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.276-283","url":null,"abstract":"The divisors of a natural number are very important for several areas of mathematics, representing a promising field in number theory. This work sought to analyze new relations involving the divisors of natural numbers, extending them to prime numbers. These are relations that may have an interesting application for counting the number of divisors of any natural number and understanding the behavior of prime numbers. They are not a primality test, but they can be a possible tool for this and could also be useful for understanding the Riemann’s zeta function that is strongly linked to the distribution of prime numbers.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48086009","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 : 2023-04-27DOI: 10.7546/nntdm.2023.29.2.260-275
Chungwu Ho, Tian-Xiao He, Peter J.-S. Shiue
As mentioned in the first part of this paper, our paper was motivated by two classical papers on the representations of integers as sums of arithmetic progressions. One of them is a paper by Sir Charles Wheatstone and the other is a paper by James Joseph Sylvester. Part I of the paper, though contained some extensions of Wheatstone’s work, was primarily devoted to extensions of Sylvester’s Theorem. In this part of the paper, we will pay more attention on the problems initiated by of Wheatstone on the representations of powers of integers as sums of arithmetic progressions and the relationships among the representations for different powers of the integer. However, a large part in this portion of the paper will be devoted to the extension of a clever method recently introduced by S. B. Junaidu, A. Laradji, and A. Umar and the problems related to the extension. This is because that this extension, not only will be our main tool for study ing the relationships of the representations of different powers of an integer, but also seems to be interesting in its own right. In the process of doing this, we need to use a few results from the first part of the paper. On the other hand, some of our results in this part will also provide certain new information on the problems studied in the first part. However, for readers who are interested primarily in the results of this part, we have repeated some basic facts from Part I of the paper so that the reader can read this part independently from the first part.
正如本文第一部分所提到的，我们的论文是由两篇关于整数表示为等差数列和的经典论文所激发的。其中一篇是查尔斯·惠斯通爵士的论文另一篇是詹姆斯·约瑟夫·西尔维斯特的论文。论文的第一部分，虽然包含了惠斯通工作的一些扩展，但主要是对西尔维斯特定理的扩展。在这一部分中，我们将更多地关注惠斯通提出的关于整数幂表示为等差数列和的问题，以及整数不同幂表示之间的关系。然而，本文这一部分的很大一部分将致力于扩展最近由S. B. Junaidu、a . Laradji和a . Umar引入的一种聪明方法，以及与此扩展相关的问题。这是因为这个扩展，不仅是我们研究整数的不同幂表示的关系的主要工具，而且它本身似乎也很有趣。在这个过程中，我们需要用到论文第一部分的一些结果。另一方面，我们在这一部分的一些结果也将为第一部分研究的问题提供一些新的信息。然而，对于主要对这一部分的结果感兴趣的读者，我们重复了论文第一部分的一些基本事实，以便读者可以独立于第一部分来阅读这一部分。
{"title":"Representations of positive integers as sums of arithmetic progressions, II","authors":"Chungwu Ho, Tian-Xiao He, Peter J.-S. Shiue","doi":"10.7546/nntdm.2023.29.2.260-275","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.260-275","url":null,"abstract":"As mentioned in the first part of this paper, our paper was motivated by two classical papers on the representations of integers as sums of arithmetic progressions. One of them is a paper by Sir Charles Wheatstone and the other is a paper by James Joseph Sylvester. Part I of the paper, though contained some extensions of Wheatstone’s work, was primarily devoted to extensions of Sylvester’s Theorem. In this part of the paper, we will pay more attention on the problems initiated by of Wheatstone on the representations of powers of integers as sums of arithmetic progressions and the relationships among the representations for different powers of the integer. However, a large part in this portion of the paper will be devoted to the extension of a clever method recently introduced by S. B. Junaidu, A. Laradji, and A. Umar and the problems related to the extension. This is because that this extension, not only will be our main tool for study ing the relationships of the representations of different powers of an integer, but also seems to be interesting in its own right. In the process of doing this, we need to use a few results from the first part of the paper. On the other hand, some of our results in this part will also provide certain new information on the problems studied in the first part. However, for readers who are interested primarily in the results of this part, we have repeated some basic facts from Part I of the paper so that the reader can read this part independently from the first part.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":"328 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136121799","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 : 2023-04-27DOI: 10.7546/nntdm.2023.29.2.241-259
Chungwu Ho, Tian-Xiao He, P. Shiue
This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.
{"title":"Representations of positive integers as sums of arithmetic progressions, I","authors":"Chungwu Ho, Tian-Xiao He, P. Shiue","doi":"10.7546/nntdm.2023.29.2.241-259","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.241-259","url":null,"abstract":"This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506814","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 : 2023-04-27DOI: 10.7546/nntdm.2023.29.2.226-240
N. Yilmaz, E. K. Çetinalp, Ö. Deveci
In this paper, we define the six different quaternion-type cyclic-Fibonacci sequences and present some properties, such as, the Cassini formula and generating function. Then, we study quaternion-type cyclic-Fibonacci sequences modulo m. Also we present the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kinds modulo m and the generating matrices of these sequences. Finally, we introduce the quaternion-type cyclic-Fibonacci sequences in finite groups. We calculate the lengths of periods for these sequences of the generalized quaternion groups and obtain quaternion-type cyclic-Fibonacci orbits of the quaternion groups Q8 and Q16 as applications of the results.
{"title":"The quaternion-type cyclic-Fibonacci sequences in groups","authors":"N. Yilmaz, E. K. Çetinalp, Ö. Deveci","doi":"10.7546/nntdm.2023.29.2.226-240","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.2.226-240","url":null,"abstract":"In this paper, we define the six different quaternion-type cyclic-Fibonacci sequences and present some properties, such as, the Cassini formula and generating function. Then, we study quaternion-type cyclic-Fibonacci sequences modulo m. Also we present the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kinds modulo m and the generating matrices of these sequences. Finally, we introduce the quaternion-type cyclic-Fibonacci sequences in finite groups. We calculate the lengths of periods for these sequences of the generalized quaternion groups and obtain quaternion-type cyclic-Fibonacci orbits of the quaternion groups Q8 and Q16 as applications of the results.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48744224","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}
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