Pub Date : 2023-03-01DOI: 10.7546/nntdm.2023.29.1.137-146
Yining Yang, Peng Yang
Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums [ H_{p}(n)=sumlimits_{substack{l_{1}+l_{2}+cdots+l_{n}=p l_{1}, l_{2}, ldots , l_{n}>0}} frac{1}{l_{1} l_{2} cdots l_{n}} ] to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 leq n leq p-6.
{"title":"Congruences for harmonic sums","authors":"Yining Yang, Peng Yang","doi":"10.7546/nntdm.2023.29.1.137-146","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.137-146","url":null,"abstract":"Zhao found a curious congruence modulo p on harmonic sums. Xia and Cai generalized his congruence to a supercongruence modulo p^2. In this paper, we improve the harmonic sums [ H_{p}(n)=sumlimits_{substack{l_{1}+l_{2}+cdots+l_{n}=p l_{1}, l_{2}, ldots , l_{n}>0}} frac{1}{l_{1} l_{2} cdots l_{n}} ] to supercongruences modulo p^3 and p^4 for odd and even where prime p>8 and 3 leq n leq p-6.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384772","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 : 2023-02-27DOI: 10.7546/nntdm.2023.29.1.78-97
Necdet Batır, A. Sofo
We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove [displaystyle sum_{k=0}^{n}frac{(-1)^{k}h_{k}}{4^{k}} {{2k} choose {k}}G_{n-k}=frac{(-1)^{n-1}}{2^{2n-1}}{{2n-2} choose {n-1}}] and [displaystyle sum_{k=1}^{infty}frac{h_{k}}{k^{2}(2k-1)4^{k}} {{2k} choose {k}}=2pi +3zeta(2)log 2-3zeta(2)-frac{7}{2}zeta(3),] where h_k=H_{2k}-dfrac{1}{2}H_{k}, G_k are Bernoulli numbers of the second kind, and zeta is the Riemann zeta function. We also give an alternate proof of the series representations for the constants log (2 pi) and gamma given by Blagouchine and Coppo.
{"title":"Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers","authors":"Necdet Batır, A. Sofo","doi":"10.7546/nntdm.2023.29.1.78-97","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.78-97","url":null,"abstract":"We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove [displaystyle sum_{k=0}^{n}frac{(-1)^{k}h_{k}}{4^{k}} {{2k} choose {k}}G_{n-k}=frac{(-1)^{n-1}}{2^{2n-1}}{{2n-2} choose {n-1}}] and [displaystyle sum_{k=1}^{infty}frac{h_{k}}{k^{2}(2k-1)4^{k}} {{2k} choose {k}}=2pi +3zeta(2)log 2-3zeta(2)-frac{7}{2}zeta(3),] where h_k=H_{2k}-dfrac{1}{2}H_{k}, G_k are Bernoulli numbers of the second kind, and zeta is the Riemann zeta function. We also give an alternate proof of the series representations for the constants log (2 pi) and gamma given by Blagouchine and Coppo.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43296373","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 : 2023-02-18DOI: 10.7546/nntdm.2023.29.1.48-61
E. Miyanohara
Let $k$ and $l$ be two multiplicatively independent positive integers and $b$ be an integer with $bge2$. Let $S$ be a finite set of integers. Nishioka proved that for any algebraic number $alpha$ with $0<|alpha|<1$ the infinite products $prod_{y=0}^{infty}(1-{alpha}^{d^{y}})$ ($d=2,3,ldots$) are algebraically independent over $mathbb{Q}$. As her result, for example, the transcendence of $prod_{y=0}^{infty}(1-frac{1}{{b}^{2^{y}}})prod_{y=0}^{infty}(1-frac{1}{{b}^{3^{y}}})$ is deduced. On the other hand, Tachiya, Amou–Väänänen investigated the certain infinite products which satisfy infinite chains of Mahler functional equation. The special case of the result of Tachiya shows that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})$ with $tau(i,y)in S$ ($1le ile k-1, yge0$) is either rational or transcendental. In this paper, we prove that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ with $tau(i,y),delta(j,y) in S$ $(1le ile k-1, 1le jle l-1, yge0)$ is either rational or transcendental. Moreover, we give sufficient conditions that $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ is transcendental.
设$k$和$l$是两个乘法独立的正整数，$b$是一个带有$bge2$的整数。设$S$是一组有限的整数。Nishioka证明了对于$0<|alpha|<1$的任何代数数$alpha$，无穷乘积$prod_｛y=0｝^｛infty｝（1-｛alpha}^｛d^｛y｝｝）$（$d=2,3，ldots$）在$mathbb｛Q｝$上是代数独立的。作为她的结果，例如，推导出$prod_｛y=0｝^｛infty｝（1-frac｛1｝｛b｝^｝2^｛y｝｝）prod_。另一方面，Tachiya，Amou–Väänänen研究了满足Mahler函数方程无穷链的某些无穷乘积。Tachiya结果的特例表明，S$（$1le ile k-1，yge0$）中$tau（i，y）的无穷乘积$prod_｛yge0｝（1+sum_｛i=1｝^｛k-1｝frac｛tau（i，y）｝｛b^｛ik^y｝｝）$要么是有理的，要么是超越的。本文证明了S$（1le ile-k-1，1le jle-1，y ge0）$要么是理性的，要么是先验的。此外，我们给出了$prod_｛yge0｝（1+sum_｛i=1｝^｛k-1｝frac｛tau（i，y）｝｛b^｛ik^y｝｝）prod_（1+ssum_｛j=1｝^{l-1｝ frac｛delta（j，y）}｛b^{jl^ y｝）$是超越的充分条件。
{"title":"Transcendental properties of the certain mix infinite products","authors":"E. Miyanohara","doi":"10.7546/nntdm.2023.29.1.48-61","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.48-61","url":null,"abstract":"Let $k$ and $l$ be two multiplicatively independent positive integers and $b$ be an integer with $bge2$. Let $S$ be a finite set of integers. Nishioka proved that for any algebraic number $alpha$ with $0<|alpha|<1$ the infinite products $prod_{y=0}^{infty}(1-{alpha}^{d^{y}})$ ($d=2,3,ldots$) are algebraically independent over $mathbb{Q}$. As her result, for example, the transcendence of $prod_{y=0}^{infty}(1-frac{1}{{b}^{2^{y}}})prod_{y=0}^{infty}(1-frac{1}{{b}^{3^{y}}})$ is deduced. On the other hand, Tachiya, Amou–Väänänen investigated the certain infinite products which satisfy infinite chains of Mahler functional equation. The special case of the result of Tachiya shows that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})$ with $tau(i,y)in S$ ($1le ile k-1, yge0$) is either rational or transcendental. In this paper, we prove that the infinite product $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ with $tau(i,y),delta(j,y) in S$ $(1le ile k-1, 1le jle l-1, yge0)$ is either rational or transcendental. Moreover, we give sufficient conditions that $prod_{yge0}(1+sum_{i=1}^{k-1} frac{tau(i,y)}{b^{ik^y}})prod_{yge0}(1+sum_{j=1}^{l-1} frac{delta(j,y)}{b^{jl^y}})$ is transcendental.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48772527","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 : 2023-02-14DOI: 10.7546/nntdm.2023.29.1.40-47
R. Vieira, Francisco Regis Vieira Al, P. Catarino
The present article deals with the study of the generalized (s,t)-Perrin sequence in its complex process. Thus, from the one-dimensional model of the generalized (s,t)-Perrin sequence, imaginary units are inserted, starting with the insertion of unit i, called two-dimensional relations. Altogether, we have the n-dimensional relationships of the generalized (s,t)-Perrin sequence.
{"title":"A study of the complexification process of the (s,t)-Perrin sequence","authors":"R. Vieira, Francisco Regis Vieira Al, P. Catarino","doi":"10.7546/nntdm.2023.29.1.40-47","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.40-47","url":null,"abstract":"The present article deals with the study of the generalized (s,t)-Perrin sequence in its complex process. Thus, from the one-dimensional model of the generalized (s,t)-Perrin sequence, imaginary units are inserted, starting with the insertion of unit i, called two-dimensional relations. Altogether, we have the n-dimensional relationships of the generalized (s,t)-Perrin sequence.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46234636","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 : 2023-02-13DOI: 10.7546/nntdm.2023.29.1.30-39
A. P. Camargo
We study the properties of two classes of functions $lambda_k$ and $tilde{lambda}_k$ that generalize the Liouville $lambda$ function, including some equivalencies between the Riemann hypothesis and some assertions about the asymptotic behavior of the summatory functions of $lambda_k$ and $tilde{lambda}_k.$ Similar results are obtained for the generalization of the Möbius function considered by Tanaka.
{"title":"Two generalizations of Liouville λ function","authors":"A. P. Camargo","doi":"10.7546/nntdm.2023.29.1.30-39","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.30-39","url":null,"abstract":"We study the properties of two classes of functions $lambda_k$ and $tilde{lambda}_k$ that generalize the Liouville $lambda$ function, including some equivalencies between the Riemann hypothesis and some assertions about the asymptotic behavior of the summatory functions of $lambda_k$ and $tilde{lambda}_k.$ Similar results are obtained for the generalization of the Möbius function considered by Tanaka.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44081319","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 : 2023-02-08DOI: 10.7546/nntdm.2023.29.1.24-29
N. Dung
We say that a positive integer $d$ is special number of degree $n$ if for every integer $m$, there exist nonzero integers $a,b,c$ such that $m=a^n+b^n-dc^n$. In this paper, we investigate some necessary conditions on $n$ for existing a special number of degree $n$.
{"title":"A note on the number $a^n+ b^n – dc^n$","authors":"N. Dung","doi":"10.7546/nntdm.2023.29.1.24-29","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.24-29","url":null,"abstract":"We say that a positive integer $d$ is special number of degree $n$ if for every integer $m$, there exist nonzero integers $a,b,c$ such that $m=a^n+b^n-dc^n$. In this paper, we investigate some necessary conditions on $n$ for existing a special number of degree $n$.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45705478","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 : 2023-02-06DOI: 10.7546/nntdm.2023.29.1.17-23
Thorranin Thansri, T. Srichan, Pinthira Tangsupphathawat
Let ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation begin{align*} sum_{nleq xatop nequiv ell pmod q}S(n), end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.
{"title":"On distribution of the number of semisimple rings of order at most x in an arithmetic progression","authors":"Thorranin Thansri, T. Srichan, Pinthira Tangsupphathawat","doi":"10.7546/nntdm.2023.29.1.17-23","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.17-23","url":null,"abstract":"Let ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation begin{align*} sum_{nleq xatop nequiv ell pmod q}S(n), end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48485426","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 : 2023-01-25DOI: 10.7546/nntdm.2023.29.1.1-16
Zehra İşbilir, M. Akyiğit, M. Tosun
In this study, we define Pauli–Leonardo quaternions by taking the coefficients of the Pauli quaternions as Leonardo numbers. We give the recurrence relation, Binet formula, generating function, exponential generating function, some special equalities, and the sum properties of these novel quaternions. In addition, we investigate the interrelations between Pauli–Leonardo quaternions and the Pauli–Fibonacci, Pauli–Lucas quaternions. Moreover, we create some algorithms that determine the terms of the Pauli–Leonardo quaternions. Finally, we generate the matrix representations of the Pauli–Leonardo quaternions and ℝ-linear transformations.
{"title":"Pauli–Leonardo quaternions","authors":"Zehra İşbilir, M. Akyiğit, M. Tosun","doi":"10.7546/nntdm.2023.29.1.1-16","DOIUrl":"https://doi.org/10.7546/nntdm.2023.29.1.1-16","url":null,"abstract":"In this study, we define Pauli–Leonardo quaternions by taking the coefficients of the Pauli quaternions as Leonardo numbers. We give the recurrence relation, Binet formula, generating function, exponential generating function, some special equalities, and the sum properties of these novel quaternions. In addition, we investigate the interrelations between Pauli–Leonardo quaternions and the Pauli–Fibonacci, Pauli–Lucas quaternions. Moreover, we create some algorithms that determine the terms of the Pauli–Leonardo quaternions. Finally, we generate the matrix representations of the Pauli–Leonardo quaternions and ℝ-linear transformations.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46364295","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 : 2022-12-05DOI: 10.7546/nntdm.2022.28.4.778-790
M. Shattuck
In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret mathcal{L}_{k,n} as the enumerator of two classes of linear colored tilings of length n. A comparable treatment is also given for the incomplete generalized Leonardo numbers. Finally, a (p,q)-generalization of mathcal{L}_{k,n} is obtained by considering the joint distribution of a pair of statistics on one of the aforementioned classes of colored tilings.
{"title":"Combinatorial proofs of identities for the generalized Leonardo numbers","authors":"M. Shattuck","doi":"10.7546/nntdm.2022.28.4.778-790","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.778-790","url":null,"abstract":"In this paper, we provide combinatorial proofs of several prior identities satisfied by the recently introduced generalized Leonardo numbers, denoted by mathcal{L}_{k,n}, as well as derive some new formulas. To do so, we interpret mathcal{L}_{k,n} as the enumerator of two classes of linear colored tilings of length n. A comparable treatment is also given for the incomplete generalized Leonardo numbers. Finally, a (p,q)-generalization of mathcal{L}_{k,n} is obtained by considering the joint distribution of a pair of statistics on one of the aforementioned classes of colored tilings.","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45528055","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 : 2022-12-05DOI: 10.7546/nntdm.2022.28.4.791-793
A. Shannon
{"title":"In Memoriam: Prof. John Turner (1928 – 2022)","authors":"A. Shannon","doi":"10.7546/nntdm.2022.28.4.791-793","DOIUrl":"https://doi.org/10.7546/nntdm.2022.28.4.791-793","url":null,"abstract":"","PeriodicalId":44060,"journal":{"name":"Notes on Number Theory and Discrete Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45543092","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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